MATHEMATICS OF COMPUTATION S 0025-5718(99)01161-8 Article electronically published on August 25, 1999

Supplement to ALMOST PERIODIC FACTORIZATION OF CERTAIN BLOCK TRIANGULAR MATRIX FUNCTIONS

ILYA M. SPITKOVSKY AND DARRYL YONG

5. Appendices

The following appendices provide explicit AP_{W} factorizations (1.2) for particular cases of the matrix (1.4), (1.5) mentioned in the body of the paper. These factorizations (as well as matrices X_{±}, G′ in the proof of Theorem 4.1) were obtained using the symbolic computation program Maple; the code used to produce them is available from the authors upon request.

To reduce the number of cases involved, we make a repeated use of the factorizations (2.8) and

G_{f^{T}} = (−J_{1}G_{+}^{−T})Λ^{−1}(G_{−}^{−T} J_{1}), (1)

obtainable from (1.2) via taking transposes and inverses. Here J_{1} = [\begin{matrix}0 & I \\ −I & 0 \end{matrix}], ^{T}

is the transposition sign, and the abbreviation X^{−T} stands for (X^{−1})^{T}. We also observe that since

G_{RfS} = diag[S^{−1}, R] G_{f} diag[S, R^{−1}],

the factorization properties of G are not changed under the transformation f ↦ RfS where R, S are invertible constant m×m matrices. This includes, in particular, elementary row and column operations applied to c_{±1} and c_{0} simultaneously.

Appendix A.

c_{−1} = [\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}] , c_{0} = [\begin{matrix}z & u & b \\ 0 & z & 0 \\ 0 & a & 0 \end{matrix}] , c_{1} = [\begin{matrix}0 & −1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] ,

and at least one of the entries a, b differs from zero.

Case 1: ab ≠ 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}−1 & 0 & e_{ν} & 0 & 0 & 0 \\ 0 & 0 & 0 & − \frac{1}{a} & e_{λ+ν} & −e_{λ} \\ \frac{z}{b} & e_{λ} & − \frac{ze_{ν} +1}{b} & − \frac{e_{α}}{ab} & 0 & 0 \\ 0 & b & 0 & 0 & ue_{ν} − e_{λ} & −u + e_{α} \\ 0 & 0 & 0 & 0 & 1 + ze_{ν} & −z \\ 0 & 0 & 0 & 0 & ae_{ν} & −a \end{matrix}] ,

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & − \frac{u}{a} \\ \frac{e_{−ν} +z}{b} & 0 & 1 & \frac{e_{−λ}}{b} & 0 & \frac{e_{−ν} −ue_{−λ}}{ab} \\ 1 & 0 & 0 & e_{−α} & 0 & − \frac{ue_{−α}}{a} \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & e_{−α} & − \frac{ze_{−α}}{a} \\ 0 & 0 & 0 & 0 & 1 & − \frac{z+e_{−ν}}{a} \end{matrix} ] ,

Λ = diag[e_{ν}, 1, e_{α}, 1, e_{−ν}, e_{−α}].

Case 2: Exactly one of the entries a, b differs from zero. Using (1) and switching the first and the second row and column of f if necessary,

we may suppose without loss of generality that a = 0, b ≠ 0. This case splits into two subcases.

Case 2a: z ≠ 0. If, in addition, ν > α, then

G_{+} = [\begin{matrix}−1 & 0 & 0 & e_{ν} & 0 & 0 \\ 0 & − \frac{1}{z} & \frac{b}{z} & 0 & \frac{ze_{λ}−e_{α}}{z} & 0 \\ \frac{z}{b} & \frac{u−e_{α}}{bz} & \frac{ze_{λ}−u+e_{α}}{z} & − \frac{1+ze_{ν}}{b} & \frac{ue_{α}−uze_{λ}−e_{2α}}{bz} & 0 \\ 0 & 0 & b & 0 & −e_{α} & 0 \\ 0 & 0 & 0 & 0 & z & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

G_{−} = [\begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ z + e_{−ν} & u & b & e_{−λ} & 1 & 0 \\ \frac{z+e_{−ν}}{b} & \frac{uz+e_{α−ν}}{bz} & 1 & \frac{e_{−λ}}{b} & \frac{e_{−ν}}{bz} & 0 \\ 1 & 0 & 0 & e_{−α} & 0 & 0 \\ 0 & \frac{z+e_{−ν}}{z} & 0 & 0 & \frac{e_{−λ}}{z} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

Λ = diag[e_{ν}, 1, 1, e_{α}, 1, e_{−λ}].

When ν < α, G can be represented as G = X_{+}G′X_{−}, where

X_{+} = [\begin{matrix}−1 & 0 & 0 & e_{ν} & 0 & 0 \\ 0 & − \frac{1}{z} & − \frac{1}{z} & 0 & be_{λ} & 0 \\ \frac{z}{b} & \frac{u−e_{α}}{bz} & \frac{u−ze_{λ}−e_{α}}{bz} & − \frac{1+ze_{ν}}{b} & e_{λ+α} − ue_{λ} & 0 \\ 0 & 0 & −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & bz & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]

is invertible in AP_{W}^{+} ,

X_{−} = [\begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ z + e_{−ν} & u & b & e_{−λ} & 1 & 0 \\ −e_{−λ} − ze_{−α} & 1 − ue_{−α} & −be_{−α} & −e_{−(λ+α)} & 0 & 0 \\ 1 & 0 & 0 & e_{−α} & 0 & 0 \\ \frac{(z+e_{−ν})^{2}}{bz} & \frac{u(z+e_{−ν})}{bz} & \frac{z+e_{−ν}}{z} & \frac{e_{−(λ+ν)}+ze_{−λ}}{bz} & \frac{e_{−ν}}{bz} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]

is invertible in AP_{W}^{−}, and

G′ = [\begin{matrix}e_{ν} & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & e_{α} & 0 & 0 & 0 \\ 0 & 0 & 0 & e_{α} & 0 & 0 \\ 0 & 0 & \frac{z+e_{−ν}}{bz} & 0 & e_{−α} & 0 \\ 0 & 0 & 0 & 0 & 0 & e_{−λ} \end{matrix}]

splits into a direct sum of diag[e_{ν}, 1, e_{α}, e_{−λ}] with a 2 × 2 matrix [ \begin{matrix}e_{α} & 0 \\ \frac{z+e_{−ν}}{bz} & e_{−α} \end{matrix}] that is

AP_{W} factorable (with zero partial AP indices) due to Theorem 1.2. This means that G′ (and therefore G itself) is also AP_{W} factorable.

Case 2b: z = 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}−1 & 0 & 0 & e_{ν} & 0 & 0 \\ 0 & −1 & 0 & 0 & e_{λ+ν} & 0 \\ 0 & \frac{u−e_{α}}{b} & e_{λ} & − \frac{1}{b} & 0 & 0 \\ 0 & 0 & b & 0 & ue_{ν} − e_{λ} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ \frac{e_{−ν}}{b} & 0 & 1 & \frac{e_{−λ}}{b} & \frac{1−ue_{−α}}{b} & 0 \\ 1 & 0 & 0 & e_{−α} & 0 & 0 \\ 0 & 1 & 0 & 0 & e_{−α} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

Λ = diag[e_{ν}, e_{ν}, 1, e_{α}, e_{−ν}, e_{−λ}].

Appendix B.

c_{0}^{(1)} = [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}] , c_{±1}^{(1)} = [\begin{matrix}x_{±} & y_{±} \\ 0 & z_{±} \end{matrix}] ,

where x_{+}z_{+} = 0, x_{−}z_{−} = 0, but at least one of the entries x_{±}, z_{±} differs from zero.

Under these conditions, the number of non-zero entries among x_{±}, z_{±} is either two or one. In the former situation, the possibilities are as follows:

(i) x_{+} = x_{−} = 0, z_{±} ≠ 0,
(ii) z_{+} = z_{−} = 0, x_{±} ≠ 0,
(iii) x_{+} = z_{−} = 0, x_{−}z_{+} ≠ 0,
(iv) z_{+} = x_{−} = 0, x_{+}z_{−} ≠ 0.

With the use of (1), (ii) can be reduced to (i) and (iii) to (iv).

If exactly one of the coefficients x_{±}, z_{±} differs from zero, the symmetries (2.8) and (1) allow us to suppose without loss of generality that x_{−} ≠ 0, x_{+} = z_{±} = 0. This leaves us with the following three cases.

Case 1: x_{+} = x_{−} = 0, z_{±} ≠ 0.

Subcase 1a: A = y_{−}z_{+} − y_{+}z_{−} ≠ 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & e_{λ+ν} − \frac{z_{−}e_{ν}}{z_{+}} & −e_{ν} & − \frac{e_{α}}{z_{−}} + \frac{z_{+}e_{λ}−z_{−}}{z_{−}A} \\ 0 & y_{+}e_{λ} + e_{ν} + y_{−} − \frac{z_{−}y_{+}}{z_{+}} & −y_{+} & − \frac{y_{+}e_{α−ν}}{z_{−}} + \frac{z_{+}+z_{+}y_{+}e_{α}}{z_{−}A} \\ 0 & z_{+}e_{λ} & −z_{+} & − \frac{z_{+}e_{α−ν}}{z_{−}} + \frac{z_{+}^{2}e_{α}}{z_{−}A} \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & \frac{z_{+}e_{−α}}{A} & − \frac{z_{+}e_{ν−α}+y_{+}z_{−}e_{−α}}{z_{−}A} \\ 0 & − \frac{z_{−}}{z_{+}} & 1 & − \frac{y_{−}z_{+}+z_{−}e_{−α}}{z_{−}z_{+}} \\ 0 & 0 & −z_{−}e_{−ν} & 1 + y_{−}e_{−ν} \end{matrix} ] ,

Λ = diag[e_{λ}, e_{−ν}, e_{−ν}, e_{ν−α}]

if ν < α, and by

G_{+} = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & (z_{−} − y_{−})e_{ν} + e_{2ν} − \frac{z_{−}e_{ν−α}}{z_{+}} & \frac{z_{−}−z_{+}e_{λ}}{A} & − \frac{e_{λ}}{z_{−}} \\ 0 & y_{+}e_{ν} − y_{−}y_{+} + e_{ν−α} + \frac{z_{−}y_{+}^{2}}{z_{+}} & − \frac{z_{+}(1+y_{+}e_{α})}{A} & − \frac{1+y_{+}e_{α}}{z_{−}} \\ 0 & z_{+}e_{ν} − A & − \frac{z_{+}^{2}e_{α}}{A} & − \frac{z_{+}e_{α}}{z_{−}} \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 − \frac{z_{−}e_{−α}}{A} & \frac{z_{+}e_{−α}}{A} & − \frac{y_{+}e_{−α}+e_{−2α}}{A} \\ 0 & − \frac{z_{−}}{z_{+}} & 1 & − \frac{y_{−}z_{+}+z_{−}e_{−α}}{z_{−}z_{+}} \\ 0 & − \frac{Az_{−}e_{α−ν}}{z_{+}} & −z_{−}e_{−ν} & 1 + \frac{z_{−}y_{+}e_{−ν}}{z_{+}} \end{matrix} ]

(with the same Λ) if ν > α.

Subcase 1b: y_{−}z_{+} −y_{+}z_{−} = 0. Using elementary row transformations, we may without loss of generality suppose that y_{+} = y_{−} = 0. Then

G_{+} = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & e_{λ} − \frac{z_{−}}{z_{+}} & − \frac{e_{λ}}{z_{−}} & −e_{ν} \\ 0 & 1 & − \frac{1}{z_{−}} & 0 \\ 0 & z_{+}e_{α} & − \frac{z_{+}e_{α}}{z_{−}} & −z_{+} \end{matrix}] ,  G_{−} = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & \frac{e_{−α}}{z_{−}} \\ 0 & 0 & −z_{−}e_{−ν} & 1 \\ 0 & − \frac{z_{−}}{z_{+}} & 1 & − \frac{e_{−α}}{z_{+}} \end{matrix} ] ,

Λ = diag[e_{λ}, 1, e_{−α}, e_{−ν}].

Case 2: x_{−} = z_{+} = 0, x_{+}z_{−} ≠ 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}e_{ν} & − \frac{y_{+}e_{λ+ν} +y_{−}e_{ν} +e_{2ν}}{x_{+}} & \frac{y_{+}}{x_{+}} & \frac{1+y_{+}e_{α}}{x_{+}z_{−}} \\ 0 & e_{λ+ν} & −1 & − \frac{e_{α}}{z_{−}} \\ x_{+} & 0 & 0 & 0 \\ 0 & z_{−} & 0 & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & \frac{y_{−}e_{−λ}+e_{−α}+y_{+}}{x_{+}} & \frac{e_{−(λ+α)}}{x_{+}} & 0 \\ 0 & 1 & 0 & \frac{e_{−α}}{z_{−}} \\ 0 & 0 & 1 & − \frac{y_{−}}{z_{−}} \\ 0 & 0 & −z_{−}e_{−ν} & 1 + y_{−}e_{−ν} \end{matrix} ] ,

Λ = diag[e_{α}, e_{−ν}, 1, e_{ν−α}].

Case 3: x_{−} ≠ 0, x_{+} = z_{+} = z_{−} = 0. Then G can be represented as G = X_{+}G′X_{−} where

X_{+} = [\begin{matrix}e_{λ+ν} & 0 & − \frac{1}{x_{−}} & − \frac{y_{−}+y_{+}e_{λ}+e_{ν}}{x_{−}} \\ 0 & 0 & 0 & 1 \\ x_{−} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ]

is invertible in AP_{W}^{+} ,

X_{−} = [\begin{matrix}1 & \frac{y_{−}}{x_{−}} & \frac{e_{−α}}{x_{−}} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}]

is invertible in AP_{W}^{−}, and

G′ = [\begin{matrix}e_{−ν} & 0 & 0 & \frac{y_{+}e_{α}+1}{x_{−}} \\ 0 & e_{−λ} & 0 & 0 \\ 0 & 0 & e_{ν} & 0 \\ 0 & 0 & 0 & e_{λ} \end{matrix} ] .

After a suitable permutation of rows and columns, the latter matrix splits into a direct sum of diag[e_{−λ}, e_{ν}] and

G_{1} = e_{α/2} [ \begin{matrix}e_{λ−α/2} & 0 \\ \frac{y_{+}e_{α/2}+e_{−α/2}}{x_{−}} & e_{−λ+α/2} \end{matrix}] .

The matrix G_{1} is AP_{W} factorable due to Theorem 1.2. This implies that G′ and the original matrix G are both AP_{W} factorable.

Of course if y_{+} = 0, then the matrix G itself satisfies the conditions of Theorem 1.2. Its explicit AP_{W} factorization in this case is given by

G_{+} = [\begin{matrix}y_{−} + e_{ν} & 0 & − \frac{1}{x_{−}} & − \frac{y_{−}e_{λ}}{x_{−}} \\ −x_{−} & 0 & 0 & e_{λ} \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \end{matrix} ] ,  G_{−} = [\begin{matrix}1 & \frac{y_{−}}{x_{−}} & \frac{e_{−α}}{x_{−}} & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ x_{−}e_{−ν} & 1 + y_{−}e_{−ν} & e_{−λ} & 0 \end{matrix}] ,

Λ = diag[e_{α}, e_{−λ}, e_{ν}, 1].

Appendix C.

c_{±1} = [\begin{matrix}a_{±} & b_{±} & d_{±} \\ f_{±} & 0 & l_{±} \\ g_{±} & 0 & 0 \end{matrix}] , c_{0} = [\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] ,

where at least on of the entries b_{±}, g_{±} differs from zero and (4.3), (4.5) hold. We sort all the possible situations by the number of non-zero entries among b_{±}, g_{±}. Situation 1: b_{±}, g_{±} are all non-zero. Then, from (4.5), l_{±} = 0. Applying elementary

row operations, we may suppose also that a_{−} = d_{−} = f_{−} = 0 (this, of course, changes a_{+}, f_{+} and d_{+}, so that equalities (4.3) do not have to hold any more).

If the (new) values of d_{+} and f_{+} happen to be zero, the matrix G splits into a direct sum of diag[e_{λ}, e_{−λ}] and a matrix satisfying Corollary 2.5. It remains to consider the following cases.

Case 1: d_{+}f_{+} ≠ 0. The AP_{W} factorization of G is then delivered by

G_{+} = [\begin{matrix}0 & e_{λ+ν} & 0 & − \frac{1}{f_{+}} & − \frac{e_{ν}}{g_{−}} & 0 \\ e_{λ} & 0 & 0 & \frac{e_{ν}}{b_{−}f_{+}} & \frac{e_{2ν}}{b_{−}g_{−}} & − \frac{1}{b_{−}} \\ − \frac{b_{+}e_{λ}+b_{−}}{d_{+}} & − \frac{a_{+}e_{λ+ν} +e_{2ν}}{d_{+}} & e_{ν} & \frac{a_{+}b_{−}−b_{+}e_{ν}}{b_{−}d_{+}f_{+}} & \frac{a_{+}b_{−}e_{ν} −b_{+}e_{2ν}}{b_{−}d_{+}g_{−}} & \frac{b_{+}}{b_{−}d_{+}} \\ 0 & 0 & d_{+} & 0 & 0 & 0 \\ 0 & f_{+}e_{λ} & 0 & 0 & − \frac{f_{+}}{g_{−}} & 0 \\ 0 & g_{+}e_{λ} + g_{−} & 0 & 0 & − \frac{g_{+}}{g_{−}} & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & \frac{g_{+}e_{ν−α}}{b_{−}f_{+}g_{−}} & − \frac{e_{ν−α}}{b_{−}g_{−}} \\ 1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ \frac{e_{−α}+a_{+}}{d_{+}} & \frac{b_{+}+b_{−}e_{−λ}}{d_{+}} & 1 & \frac{e_{−(λ+α)}}{d_{+}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{+}+g_{−}e_{−λ}}{f_{+}} & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{matrix} ] ,

Λ = diag[1, e_{−ν}, e_{α}, e_{−α}, 1, e_{ν}]

if ν < α, and

G_{+} = [\begin{matrix}0 & e_{λ+ν} & 0 & − \frac{1}{f_{+}} & − \frac{e_{ν}}{g_{−}} & 0 \\ e_{λ} & 0 & 0 & \frac{e_{ν}}{b_{−}f_{+}} & 0 & − \frac{1}{b_{−}} \\ − \frac{b_{−}+b_{+}e_{λ}}{d_{+}} & − \frac{a_{+}e_{λ+ν} +e_{2ν}}{d_{+}} & e_{ν} & \frac{a_{+}b_{−}−b_{+}e_{ν}}{b_{−}d_{+}f_{+}} & \frac{a_{+}e_{ν} +e_{ν−α}}{d_{+}g_{−}} & \frac{b_{+}}{b_{−}d_{+}} \\ 0 & 0 & d_{+} & 0 & 0 & 0 \\ 0 & f_{+}e_{λ} & 0 & 0 & − \frac{f_{+}}{g_{−}} & 0 \\ 0 & g_{+}e_{λ} + g_{−} & 0 & 0 & − \frac{g_{+}}{g_{−}} & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & − \frac{e_{−2α}}{b_{−}f_{+}} & 0 \\ 1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ \frac{e_{−α}+a_{+}}{d_{+}} & \frac{b_{+}+b_{−}e_{−λ}}{d_{+}} & 1 & \frac{e_{−(λ+α)}}{d_{+}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{+}+g_{−}e_{−λ}}{f_{+}} & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \end{matrix} ]

(with the same Λ) if ν > α.

Case 2: Exactly one of d_{+}, f_{+} differs from zero. Using (1), we may without loss of generality suppose that d_{+} = 0, f_{+} ≠ 0. The AP_{W} factorization of G is then delivered by

G_{+} = [\begin{matrix}e_{λ+ν} & 0 & 0 & − \frac{e_{ν}}{g_{−}} & 0 & − \frac{1}{f_{+}} \\ − \frac{e_{2ν}}{b_{+}} & \frac{b_{+}e_{λ}−b_{−}}{b_{+}} & 0 & \frac{e_{2ν}}{b_{−}g_{−}} & − \frac{e_{ν}}{b_{−}} & \frac{a_{+}}{b_{+}f_{+}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ a_{+}e_{λ} & b_{+}e_{α} & 0 & \frac{b_{+}e_{ν} −a_{+}b_{−}}{b_{−}g_{−}} & − \frac{b_{+}}{b_{−}} & 0 \\ f_{+}e_{λ} & 0 & 0 & − \frac{f_{+}}{g_{−}} & 0 & 0 \\ g_{+}e_{λ} + g_{−} & 0 & 0 & − \frac{g_{+}}{g_{−}} & 0 & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ 0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & \frac{g_{+}e_{ν−α}}{b_{−}f_{+}g_{−}} & − \frac{e_{ν−α}}{b_{−}g_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{−}e_{−λ}+g_{+}}{f_{+}} & 1 \\ − \frac{b_{−}}{b_{+}} & − \frac{b_{−}^{2}e_{−ν}}{b_{+}} & 0 & \frac{b_{+}−b_{−}e_{−λ}}{b_{+}} & \frac{a_{+}b_{−}e_{−λ}−b_{+}e_{−α}}{b_{+}f_{+}} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix} ] ,

Λ = diag[e_{−ν}, 1, e_{λ}, 1, 1, e_{−α}]

if ν < α, and

G_{+} = [\begin{matrix}e_{λ+ν} & 0 & 0 & − \frac{e_{ν}}{g_{−}} & 0 & − \frac{1}{f_{+}} \\ − \frac{e_{2ν}}{b_{+}} & \frac{b_{+}e_{λ}−b_{−}}{b_{+}} & 0 & \frac{e_{ν−α}}{b_{+}g_{−}} & − \frac{e_{ν}}{b_{−}} & \frac{a_{+}}{b_{+}f_{+}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ a_{+}e_{λ} & b_{+}e_{α} & 0 & − \frac{a_{+}}{g_{−}} & − \frac{b_{+}}{b_{−}} & 0 \\ f_{+}e_{λ} & 0 & 0 & − \frac{f_{+}}{g_{−}} & 0 & 0 \\ g_{+}e_{λ} + g_{−} & 0 & 0 & − \frac{g_{+}}{g_{−}} & 0 & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ 0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & − \frac{e_{−2α}}{b_{−}f_{+}} & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{−}e_{−λ}+g_{+}}{f_{+}} & 1 \\ − \frac{b_{−}}{b_{+}} & − \frac{b_{−}^{2}e_{−ν}}{b_{+}} & 0 & \frac{b_{+}−b_{−}e_{−λ}}{b_{+}} & \frac{a_{+}b_{−}e_{−λ}−b_{+}e_{−α}}{b_{+}f_{+}} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix} ]

(with the same Λ) if ν > α.

Situation 2: Exactly one of the numbers b_{±}, g_{±} is zero.

Using the symmetries (2.8) and (1), we may suppose that b_{−} = 0, b_{+}, g_{±} ≠ 0. From (4.5), (4.3) it follows that l_{±} = 0. Applying elementary row and column operations, we may then ensure that f_{−} = a_{±} = d_{+} = 0; of course, the latter transformations change the value of f_{+}. If the (new) f_{+} and d_{−} both equal zero, then (as in Situation 1) G splits into a direct sum of a diagonal matrix with a matrix AP_{W} factorable due to Theorem 1.2. The remaining three possibilities are covered by the following cases.

Case 3: d_{−}f_{+} ≠ 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}0 & − \frac{e_{ν}}{g_{−}} & 0 & − \frac{1}{f_{+}} & e_{λ+ν} & 0 \\ 0 & 0 & e_{ν} & 0 & − \frac{e_{2ν}}{b_{+}} & − \frac{d_{−}}{b_{+}} \\ − \frac{1}{d_{−}} & \frac{e_{2ν}}{d_{−}g_{−}} & 0 & \frac{e_{ν}}{d_{−}f_{+}} & 0 & e_{λ} \\ 0 & 0 & b_{+} & 0 & 0 & 0 \\ 0 & − \frac{f_{+}}{g_{−}} & 0 & 0 & f_{+}e_{λ} & 0 \\ 0 & − \frac{g_{+}}{g_{−}} & 0 & 0 & g_{+}e_{λ} + g_{−} & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{+}+g_{−}e_{−λ}}{f_{+}} & 1 \\ \frac{e_{−α}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 1 & \frac{e_{−α}}{d_{−}} & \frac{g_{+}e_{ν−α}}{d_{−}f_{+}g_{−}} & − \frac{e_{ν−α}}{d_{−}g_{−}} \end{matrix} ] ,

Λ = diag[e_{ν}, 1, e_{α}, e_{−α}, e_{−ν}, 1]

if ν < α, and

G_{+} = [\begin{matrix}0 & − \frac{e_{ν}}{g_{−}} & 0 & − \frac{1}{f_{+}} & e_{λ+ν} & 0 \\ 0 & \frac{e_{ν−α}}{b_{+}g_{−}} & e_{ν} & 0 & − \frac{e_{2ν}}{b_{+}} & − \frac{d_{−}}{b_{+}} \\ − \frac{1}{d_{−}} & 0 & 0 & \frac{e_{ν}}{d_{−}f_{+}} & 0 & e_{λ} \\ 0 & 0 & b_{+} & 0 & 0 & 0 \\ 0 & − \frac{f_{+}}{g_{−}} & 0 & 0 & f_{+}e_{λ} & 0 \\ 0 & − \frac{g_{+}}{g_{−}} & 0 & 0 & g_{+}e_{λ} + g_{−} & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{+}+g_{−}e_{−λ}}{f_{+}} & 1 \\ \frac{e_{−α}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 1 & \frac{e_{−α}}{d_{−}} & − \frac{e_{−2α}}{d_{−}f_{+}} & 0 \end{matrix} ]

(with the same Λ) if ν > α.

Case 4: d_{−} = 0, f_{+} ≠ 0. The AP_{W} factorization of G is delivered by

G_{+} = [\begin{matrix}e_{λ+ν} & 0 & 0 & − \frac{1}{f_{+}} & −e_{ν} & − \frac{e_{α}}{g_{−}} \\ 0 & e_{ν} & 0 & 0 & 0 & \frac{1}{b_{+}g_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ e_{ν} & b_{+} & 0 & 0 & 0 & 0 \\ f_{+}e_{λ} & 0 & 0 & 0 & −f_{+} & − \frac{f_{+}e_{α−ν}}{g_{−}} \\ g_{+}e_{λ} + g_{−} & 0 & 0 & 0 & −g_{+} & − \frac{g_{+}e_{α−ν}}{g_{−}} \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ 0 & 1 & 0 & \frac{e_{−(λ+α)}}{b_{+}} & \frac{g_{+}e_{−2α}}{b_{+}f_{+}g_{−}} & − \frac{e_{−2α}}{b_{+}g_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & − \frac{e_{−α}}{f_{+}} & 0 \\ 0 & 0 & 0 & −g_{−}e_{−ν} & − \frac{g_{+}}{f_{+}} & 1 \end{matrix} ] ,

Λ = diag[e_{−ν}, e_{α}, e_{λ}, e_{−α}, 1, e_{−α}]

if ν < α, and

G_{+} = [\begin{matrix}e_{λ+ν} & 0 & 0 & − \frac{1}{f_{+}} & − \frac{e_{ν}}{g_{−}} & 0 \\ 0 & e_{ν} & 0 & 0 & \frac{e_{ν−α}}{b_{+}g_{−}} & − \frac{1}{b_{+}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ e_{ν} & b_{+} & 0 & 0 & 0 & 0 \\ f_{+}e_{λ} & 0 & 0 & 0 & − \frac{f_{+}}{g_{−}} & 0 \\ g_{+}e_{λ} + g_{−} & 0 & 0 & 0 & − \frac{g_{+}}{g_{−}} & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ 0 & 1 & 0 & \frac{e_{−(λ+α)}}{b_{+}} & \frac{g_{+}e_{−2α}}{b_{+}f_{+}g_{−}} & − \frac{e_{−2α}}{b_{+}g_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{+}+g_{−}e_{−λ}}{f_{+}} & 1 \\ 0 & 0 & 0 & 1 & − \frac{e_{−α}}{f_{+}} & 0 \end{matrix} ] ,

Λ = diag[e_{−ν}, e_{α}, e_{λ}, e_{−α}, 1, e_{−α}]

if ν > α.

Case 5: d_{−} ≠ 0, f_{+} = 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}0 & 0 & 0 & e_{λ} − \frac{g_{−}}{g_{+}} & 0 & − \frac{e_{ν}}{g_{−}} \\ 0 & − \frac{d_{−}}{b_{+}} & e_{ν} & − \frac{e_{ν}}{b_{+}} & 0 & 0 \\ − \frac{1}{d_{−}} & e_{λ} & 0 & \frac{g_{−}e_{ν}}{d_{−}g_{+}} & 0 & \frac{e_{2ν}}{d_{−}g_{−}} \\ 0 & 0 & b_{+} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & g_{+}e_{α} & 0 & − \frac{g_{+}}{g_{−}} \end{matrix} ] ,

G_{−} = [\begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{e_{−α}}{d_{−}} & 0 & − \frac{e_{ν−α}}{d_{−}g_{−}} \\ \frac{e_{−α}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ − \frac{g_{−}^{2}e_{−ν}}{g_{+}} & 0 & 0 & 0 & 0 & 1 − \frac{g_{−}e_{−λ}}{g_{+}} \end{matrix} ] ,

Λ = diag[e_{ν}, 1, e_{α}, 1, e_{−λ}, 1] if ν < α, and

G_{+} = [\begin{matrix}0 & 0 & 0 & e_{λ} − \frac{g_{−}}{g_{+}} & 0 & − \frac{e_{ν}}{g_{−}} \\ 0 & − \frac{d_{−}}{b_{+}} & e_{ν} & − \frac{e_{ν}}{b_{+}} & 0 & \frac{e_{ν−α}}{b_{+}g_{−}} \\ − \frac{1}{d_{−}} & e_{λ} & 0 & \frac{g_{−}e_{ν}}{d_{−}g_{+}} & 0 & 0 \\ 0 & 0 & b_{+} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & g_{+}e_{α} & 0 & − \frac{g_{+}}{g_{−}} \end{matrix} ] ,

G_{−} = [\begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ − \frac{g_{−}e_{−α}}{d_{−}g_{+}} & 0 & 1 & \frac{e_{−α}}{d_{−}} & 0 & − \frac{e_{−2α}}{d_{−}g_{+}} \\ \frac{e_{−α}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ − \frac{g_{−}^{2}e_{−ν}}{g_{+}} & 0 & 0 & 0 & 0 & 1 − \frac{g_{−}e_{−λ}}{g_{+}} \end{matrix} ]

(with the same Λ) if ν > α.

We now pass to the situations in which exactly two of the coefficients b_{±}, g_{±} differ from zero. There are six such situations overall, but due to the symmetries (2.8) and (1), half of them can be dropped. The remaining three possibilities are considered in Situations 3–5 below.

Situation 3: b_{+} = g_{+} = 0, b_{−}g_{−} ≠ 0.

From (4.5), (4.3) it follows that l_{±} = 0. Using elementary row and column operations, we may without loss of generality suppose that a_{+} = f_{+} = d_{+} = 0. This does not change the values of a_{−}, d_{−}, and the further classification of possible cases is based on the zero/non-zero pattern of d_{−}, f_{−}. If both of them equal zero, the matrix G splits into a direct sum of a diagonal matrix and a matrix satisfying Corollary 2.5 (compare with a similar case in Situation 1). The remaining options are covered by Cases 6 and 7.

Case 6: d_{−}f_{−} ≠ 0. The AP_{W} factorization of G is delivered by

G_{+} = [\begin{matrix}0 & − \frac{1}{g_{+}} & e_{λ+ν} & − \frac{e_{ν}}{f_{−}} & 0 & 0 \\ 0 & 0 & 0 & 0 & e_{ν} & − \frac{d_{−}}{b_{+}} \\ − \frac{1}{d_{−}} & \frac{e_{ν} +a_{−}}{d_{−}g_{+}} & − \frac{a_{−}e_{λ+ν}}{d_{−}} & \frac{e_{2ν} +a_{−}e_{ν}}{d_{−}f_{−}} & 0 & e_{λ} \\ 0 & 0 & e_{ν} & 0 & b_{+} & 0 \\ 0 & 0 & f_{−} & 0 & 0 & 0 \\ 0 & 0 & g_{+}e_{λ} & − \frac{g_{+}}{f_{−}} & 0 & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & \frac{e_{−α}}{f_{−}} & 0 \\ 0 & 0 & 0 & 0 & 1 & − \frac{f_{−}e_{−λ}}{g_{+}} \\ \frac{a_{−}e_{−λ}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & − \frac{e_{−2α}}{b_{+}f_{−}} & 0 \\ \frac{a_{−}}{d_{−}} & 0 & 1 & \frac{e_{−α}}{d_{−}} & − \frac{e_{ν−α}}{d_{−}f_{−}} & 0 \end{matrix} ] ,

Λ = diag[e_{ν}, e_{−α}, e_{−ν}, 1, e_{α}, 1]

if ν < α, and by

G_{+} = [\begin{matrix}0 & − \frac{1}{g_{+}} & e_{λ+ν} & − \frac{e_{ν}}{f_{−}} & 0 & 0 \\ 0 & 0 & 0 & \frac{e_{ν−α}}{b_{+}f_{−}} & e_{ν} & − \frac{d_{−}}{b_{+}} \\ − \frac{1}{d_{−}} & \frac{e_{ν} +a_{−}}{d_{−}g_{+}} & − \frac{a_{−}e_{λ+ν}}{d_{−}} & \frac{a_{−}e_{ν}}{d_{−}f_{−}} & 0 & e_{λ} \\ 0 & 0 & e_{ν} & 0 & b_{+} & 0 \\ 0 & 0 & f_{−} & 0 & 0 & 0 \\ 0 & 0 & g_{+}e_{λ} & − \frac{g_{+}}{f_{−}} & 0 & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & \frac{e_{−α}}{f_{−}} & 0 \\ 0 & 0 & 0 & 0 & 1 & − \frac{f_{−}e_{−λ}}{g_{+}} \\ \frac{a_{−}e_{−λ}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & − \frac{e_{−2α}}{b_{+}f_{−}} & 0 \\ \frac{a_{−}}{d_{−}} & 0 & 1 & \frac{e_{−α}}{d_{−}} & 0 & − \frac{e_{−2α}}{d_{−}g_{+}} \end{matrix} ]

(with the same Λ) if ν > α.

Case 7: Exactly one of the entries d_{−}, f_{−} differs from zero.

Using the symmetry (1), we may without loss of generality suppose that d_{−} = 0, f_{−} ≠ 0. Then the AP_{W} factorization of G is delivered by

G_{+} = [\begin{matrix}e_{λ+ν} & 0 & 0 & −e_{ν} & − \frac{1}{g_{+}} & − \frac{e_{α}}{f_{−}} \\ 0 & e_{ν} & 0 & 0 & 0 & \frac{1}{b_{+}f_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ e_{ν} + a_{−} & b_{+} & 0 & 0 & 0 & 0 \\ f_{−} & 0 & 0 & 0 & 0 & 0 \\ g_{+}e_{λ} & 0 & 0 & −g_{+} & 0 & − \frac{g_{+}e_{α−ν}}{f_{−}} \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 & \frac{e_{−α}}{f_{−}} & 0 \\ 0 & 1 & 0 & \frac{e_{−(λ+α)}}{b_{+}} & − \frac{a_{−}e_{−(λ+α)}+e_{−2α}}{b_{+}f_{−}} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & − \frac{a_{−}}{f_{−}} & − \frac{e_{−α}}{g_{+}} \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & −f_{−}e_{−ν} & a_{−}e_{−ν} + 1 & 0 \end{matrix} ] ,

Λ = diag[e_{−ν}, e_{α}, e_{λ}, e_{−ν}, e_{−α}, e_{ν−α}]

if ν < α, and

G_{+} = [\begin{matrix}e_{λ+ν} & 0 & 0 & 0 & − \frac{1}{g_{+}} & − \frac{e_{α}}{f_{−}} \\ 0 & e_{ν} & 0 & − \frac{1}{b_{+}} & 0 & \frac{e_{ν−α}}{b_{+}f_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ e_{ν} + a_{−} & b_{+} & 0 & 0 & 0 & 0 \\ f_{−} & 0 & 0 & 0 & 0 & 0 \\ g_{+}e_{λ} & 0 & 0 & 0 & 0 & − \frac{g_{+}}{f_{−}} \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 & \frac{e_{−α}}{f_{−}} & 0 \\ 0 & 1 & 0 & − \frac{e_{−(λ+α)}}{b_{+}} & − \frac{a_{−}e_{−(λ+α)}+e_{−2α}}{b_{+}f_{−}} & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & − \frac{a_{−}}{f_{−}} & − \frac{e_{−α}}{g_{+}} \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & − \frac{f_{−}e_{−λ}}{g_{+}} \end{matrix} ] ,

Λ = diag[e_{−ν}, e_{α}, e_{λ}, e_{−α}, e_{−α}, 1]

if ν > α.

Situation 4: b_{±} = 0, g_{±} ≠ 0.

Conditions (4.5) and the second equality of (4.3) are then satisfied automatically. The other two equalities of (4.3) take the form

l_{+}g_{−} = l_{−}g_{+}, d_{+}g_{−} = d_{−}g_{+}. (2)

Due to the first condition in (2), either l_{+} = l_{−} = 0 or l_{±} ≠ 0. We now consider these two cases separately.

Case 8: l_{±} ≠ 0.

Using elementary row operations, we may without loss of generality suppose that f_{−} = a_{−} = d_{−} = 0; due to (2), we will simultaneously achieve that d_{+} = 0. Applying elementary column operations, we may then assure that f_{+} = 0. There are now two subcases, depending on the new value of a_{+}.

Subcase 8a: a_{+} ≠ 0. The desired AP_{W} factorization is given by

G_{+} = [\begin{matrix}0 & 0 & e_{λ+ν} & \frac{g_{−}−g_{+}e_{λ}−a_{+}g_{−}e_{α}}{a_{+}g_{−}^{2}} & 0 & \frac{g_{+}e_{λ+ν} −g_{−}e_{ν}}{g_{−}} \\ 0 & 1 & 0 & 0 & 0 & 0 \\ e_{λ} − \frac{l_{−}}{k_{+}} & 0 & 0 & 0 & − \frac{e_{ν}}{l_{−}} & 0 \\ 0 & 0 & e_{ν} + a_{+}e_{λ} & − \frac{a_{+}g_{+}e_{α}+g_{+}+a_{+}^{2}g_{−}e_{α−ν}}{a_{+}g_{−}^{2}} & 0 & \frac{a_{+}g_{+}e_{λ}+g_{+}e_{ν}}{g_{−}} − a_{+} \\ l_{+}e_{α} & 0 & 0 & 0 & − \frac{l_{+}}{l_{−}} & 0 \\ 0 & 0 & g_{+}e_{λ} + g_{−} & − \frac{g_{+}(a_{+}g_{−}e_{α−ν} +g_{+}e_{α})}{a_{+}g_{−}^{2}} & 0 & \frac{g_{+}^{2}e_{λ}}{g_{−}} \end{matrix}] ,

G_{−} = [\begin{matrix}0 & 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} & 0 \\ 0 & 1 & & 0 & 0 & 0 \\ 1 & 0 & 0 & − \frac{g_{+}}{g_{−}} & 0 & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 0 & −g_{−}e_{−ν} & 0 & 1 \\ 0 & 0 & − \frac{l_{−}^{2}e_{−ν}}{l_{+}} & 0 & 1 − \frac{l_{−}e_{−λ}}{l_{+}} & 0 \\ 0 & 0 & 0 & 1 − \frac{e_{−α}}{a_{+}} & 0 & \frac{e_{ν−α}}{a_{+}g_{−}} \end{matrix}  ] ,

Λ = diag[1, e_{λ}, e_{−ν}, e_{ν−α}, 1, e_{−ν}]

if ν < α, and by

G_{+} = [ \begin{matrix}\frac{g_{+}e_{λ}−g_{−}}{a_{+}g_{−}} & 0 & e_{2ν} + \frac{g_{−}(a_{+}e_{ν} −e_{ν−α})}{g_{+}} & 0 & 0 & − \frac{e_{λ}}{g_{−}} \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & e_{λ} − \frac{l_{−}}{l_{+}} & − \frac{e_{ν}}{l_{−}} & 0 \\ \frac{g_{+}(1+a_{+}e_{α})}{a_{+}g_{−}} & 0 & a_{+}e_{ν} + e_{ν−α} + \frac{a_{+}^{2}g_{−}}{g_{+}} & 0 & 0 & − \frac{1+a_{+}e_{α}}{g_{−}} \\ 0 & 0 & 0 & l_{+}e_{α} & − \frac{l_{+}}{l_{−}} & 0 \\ \frac{g_{+}^{2}e_{α}}{a_{+}g_{−}} & 0 & g_{+}e_{ν} + a_{+}g_{−} & 0 & 0 & − \frac{g_{+}e_{α}}{g_{−}} \end{matrix} ] ,

G_{−} = [ \begin{matrix}− \frac{g_{−}}{g_{+}} & 0 & 0 & 1 & 0 & − \frac{e_{−α}}{g_{+}} \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 + \frac{e_{−α}}{a_{+}} & 0 & 0 & − \frac{g_{+}e_{−α}}{a_{+}g_{−}} & 0 & \frac{a_{+}e_{−α}+e_{−2α}}{a_{+}g_{−}} \\ 0 & 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} & 0 \\ 0 & 0 & − \frac{l_{−}^{2}e_{−ν}}{l_{+}} & 0 & 1 − \frac{l_{−}e_{−λ}}{l_{+}} & 0 \\ \frac{a_{+}g_{−}^{2}e_{α−ν}}{g_{+}} & 0 & 0 & −g_{−}e_{−ν} & 0 & 1 + \frac{a_{+}g_{−}e_{−ν}}{g_{+}} \end{matrix} ] ,

Λ = diag[e_{−α}, e_{λ}, e_{α−ν}, 1, 1, e_{−α}]

if ν > α.

Subcase 8b: a_{+} = 0. The desired AP_{W} factorization is given by

G_{+} = [\begin{matrix}0 & 0 & − \frac{e_{λ}}{g_{−}} & \frac{g_{+}e_{λ}}{g_{−}} − 1 & 0 & e_{λ+ν} \\ 0 & 1 & 0 & 0 & 0 & 0 \\ e_{λ} − \frac{l_{−}}{l_{+}} & 0 & 0 & \frac{f_{+}e_{λ}}{l_{−}} & − \frac{e_{ν}}{l_{−}} & 0 \\ 0 & 0 & − \frac{1}{g_{−}} & \frac{g_{+}}{g_{−}} & 0 & e_{ν} \\ l_{+}e_{α} & 0 & − \frac{f_{+}e_{α}}{g_{−}} & \frac{f_{+}e_{α}(g_{+}l_{−}+g_{−}l_{+})}{g_{−}l_{−}} & − \frac{l_{+}}{l_{−}} & f_{+}e_{λ} \\ 0 & 0 & − \frac{g_{+}e_{α}}{g_{−}} & \frac{g_{+}^{2}e_{α}}{g_{−}} & 0 & g_{+}e_{λ} + g_{−} \end{matrix}] ,

G_{−} = [\begin{matrix}0 & 0 & 1 & − \frac{f_{+}}{l_{−}} & \frac{e_{−α}}{l_{−}} & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −g_{−}e_{−ν} & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & − \frac{l_{−}^{2}e_{−ν}}{l_{+}} & \frac{f_{+}l_{−}e_{−ν}}{l_{+}} & 1 − \frac{l_{−}e_{−λ}}{l_{+}} & 0 \\ 1 & 0 & 0 & − \frac{g_{+}}{g_{−}} & 0 & \frac{e_{−α}}{g_{−}} \end{matrix} ] ,

Λ = diag[1, e_{λ}, e_{−α}, 1, 1, e_{−ν}].

Case 9: l_{+} = l_{−} = 0.

Using elementary row operations, we can make a_{−} = f_{−} = 0. If in addition f_{+} = 0, then G splits into a direct sum of a diagonal matrix diag[e_{λ}, e_{−λ}] and another matrix G_{1} of the type (1.4), with m = 2 and

c_{±1} = [\begin{matrix}d_{±} & 0 \\ 0 & g_{±} \end{matrix}] , c_{0} = [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}] .

Due to the second equation in (2), either d_{±} ≠ 0 or d_{+} = d_{−} = 0. In the former case, G_{1} satisfies the conditions of Corollary 2.5. The latter case is covered by Case 1 of Appendix B. Either way, G_{1} is AP_{W} factorable, which implies AP_{W} factorability of G.

It remains to consider the case f_{+} ≠ 0. Applying another elementary row operation, we may suppose that a_{+} = 0. Due to (2), it is still true that either d_{+} = d_{−} = 0 or d_{±} ≠ 0. According to this, the following subcases are in order.

Subcase 9a: d_{+} = d_{−} = 0. The desired AP_{W} factorization is given by

G_{+} = [\begin{matrix}0 & 0 & − \frac{e_{λ}}{g_{−}} & \frac{g_{+}e_{λ}−g_{−}}{f_{+}g_{−}} & \frac{g_{+}e_{λ+ν}}{g_{−}} − e_{ν} & e_{λ+ν} \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & − \frac{1}{g_{−}} & \frac{g_{+}}{f_{+}g_{−}} & \frac{g_{+}e_{ν}}{g_{−}} & e_{ν} \\ 0 & 0 & − \frac{f_{+}e_{α}}{g_{−}} & \frac{g_{+}e_{α}}{g_{−}} & \frac{f_{+}(g_{+}e_{λ}−g_{−})}{g_{−}} & f_{+}e_{λ} \\ 0 & 0 & − \frac{g_{+}e_{α}}{g_{−}} & \frac{g_{+}^{2}e_{α}}{f_{+}g_{−}} & \frac{g_{+}^{2}e_{λ}}{g_{−}} & g_{+}e_{λ} + g_{−} \end{matrix}] ,

G_{−} = [\begin{matrix}0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −g_{−}e_{−ν} & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & − \frac{e_{−α}}{f_{+}} & 0 \\ 1 & 0 & 0 & − \frac{g_{+}}{g_{−}} & 0 & \frac{e_{−α}}{g_{−}} \end{matrix} ] ,

Λ = diag[e_{λ}, e_{λ}, e_{−α}, e_{−α}, e_{−ν}, e_{−ν}].

Subcase 9b: d_{±} ≠ 0. The desired AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}0 & 0 & e_{λ+ν} & \frac{g_{+}e_{λ}−g_{−}}{f_{+}g_{−}} & − \frac{e_{ν}}{g_{−}} & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ − \frac{e_{ν}}{d_{−}} & 0 & − \frac{e_{2ν}}{d_{+}} & \frac{e_{ν} (g_{−}d_{+}−g_{+}d_{−})}{d_{+}d_{−}f_{+}g_{−}} & \frac{e_{2ν}}{d_{−}g_{−}} & e_{λ} − \frac{d_{−}}{d_{+}} \\ − \frac{d_{+}}{d_{−}} & 0 & 0 & \frac{d_{+}}{d_{−}f_{+}} & \frac{d_{+}e_{ν}}{d_{−}g_{−}} & d_{+}e_{α} \\ 0 & 0 & f_{+}e_{λ} & \frac{g_{+}e_{α}}{g_{−}} & − \frac{f_{+}}{g_{−}} & 0 \\ 0 & 0 & g_{+}e_{λ} + g_{−} & \frac{g_{+}^{2}e_{α}}{f_{+}e_{−}} & − \frac{g_{+}}{g_{−}} & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}− \frac{d_{−}}{d_{+}} & 0 & − \frac{d_{−}^{2}e_{−ν}}{d_{+}} & 1 − \frac{d_{−}e_{−λ}}{d_{+}} & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{−}e_{−λ}}{f_{+}} & 1 \\ 0 & 0 & 1 & \frac{e_{−α}}{d_{−}} & 0 & − \frac{e_{ν−α}}{d_{−}g_{−}} \end{matrix} ] ,

Λ = diag[1, e_{λ}, e_{−ν}, e_{−α}, 1, 1]

if ν < α, and by

G_{+} = [\begin{matrix}0 & 0 & 0 & − \frac{e_{ν}}{g_{−}} & e_{λ+ν} − \frac{g_{−}e_{ν}}{g_{+}} & \frac{g_{+}e_{λ}−g_{−}}{f_{+}g_{−}} \\ 0 & 1 & 0 & 0 & 0 & 0 \\ − \frac{e_{ν}}{d_{−}} & 0 & e_{λ} − \frac{d_{−}}{d_{+}} & \frac{e_{ν−α}}{d_{+}g_{−}} & \frac{e_{2ν} (g_{−}d_{+}−g_{+}d_{−})}{d_{+}d_{−}g_{+}} & \frac{e_{ν} (g_{−}d_{+}−g_{+}d_{−})}{d_{+}d_{−}f_{+}g_{−}} \\ − \frac{d_{+}}{d_{−}} & 0 & d_{+}e_{α} & 0 & \frac{d_{+}g_{−}e_{ν}}{d_{−}g_{+}} & \frac{d_{+}}{d_{−}f_{+}} \\ 0 & 0 & 0 & − \frac{f_{+}}{g_{−}} & f_{+}e_{λ} − \frac{f_{+}g_{−}}{g_{+}} & \frac{g_{+}e_{α}}{g_{−}} \\ 0 & 0 & 0 & − \frac{g_{+}}{g_{−}} & g_{+}e_{λ} & \frac{g_{+}^{2}e_{α}}{f_{+}g_{−}} \end{matrix} ] ,

G_{−} = [\begin{matrix}− \frac{d_{−}}{d_{+}} & 0 & − \frac{d_{−}^{2}e_{−ν}}{d_{+}} & 1 − \frac{d_{−}e_{−λ}}{d_{+}} & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ − \frac{g_{−}e_{−α}}{d_{−}g_{+}} & 0 & 1 & \frac{e_{−α}}{d_{−}} & 0 & − \frac{e_{−2α}}{d_{−}g_{+}} \\ − \frac{g_{−}^{2}e_{−ν}}{g_{+}} & 0 & 0 & 0 & 0 & 1 − \frac{g_{−}e_{−λ}}{g_{+}} \\ 1 & 0 & 0 & 0 & − \frac{g_{+}e_{−α}}{f_{+}g_{−}} & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 0 & 0 & 1 & 0 \end{matrix} ] ,

Λ = diag[1, e_{λ}, 1, 1, e_{−ν}, e_{−α}]

if ν > α.

Situation 5: b_{−} = g_{+} = 0, b_{+}g_{−} ≠ 0.

From the first two equations of (4.3) it follows that l_{±} = 0. Applying elementary row and column operations, we may suppose without loss of generality that a_{±} = d_{+} = f_{−} = 0; of course, this leads to a change in the value of f_{+}. If the new f_{+} and d_{−} equal zero, then the matrix G splits into a direct sum of a diagonal matrix diag[e_{λ}, e_{−λ}] and a matrix G_{1} of the type (1.4), (1.5) with m = 2 and

c_{−1} = [\begin{matrix}0 & 0 \\ 0 & g_{−} \end{matrix}] , c_{1} = [\begin{matrix}b_{+} & 0 \\ 0 & 0 \end{matrix}] , c_{0} = [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}] .

The latter matrix is AP_{W} factorable due to Case 2 of Appendix B. Hence, the original matrix G is also AP_{W} factorable.

The remaining three possibilities (in terms of zero/non-zero pattern of the entries d_{−}, f_{+}) are covered by Cases 10–12.

Case 10: d_{−}f_{+} ≠ 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}0 & − \frac{e_{ν}}{g_{−}} & 0 & − \frac{1}{f_{+}} & e_{λ+ν} & 0 \\ 0 & 0 & e_{ν} & 0 & − \frac{e_{2ν}}{b_{+}} & − \frac{d_{−}}{b_{+}} \\ − \frac{1}{d_{−}} & \frac{e_{2ν}}{d_{−}g_{−}} & 0 & \frac{e_{ν}}{d_{−}f_{+}} & 0 & e_{λ} \\ 0 & 0 & b_{+} & 0 & 0 & 0 \\ 0 & − \frac{f_{+}}{g_{−}} & 0 & 0 & f_{+}e_{λ} & 0 \\ 0 & 0 & 0 & 0 & g_{−} & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{−}e_{−λ}}{f_{+}} & 1 \\ \frac{e_{−α}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 1 & \frac{e_{−α}}{d_{−}} & 0 & − \frac{e_{ν−α}}{d_{−}g_{−}} \end{matrix} ] ,

Λ = diag[e_{ν}, 1, e_{α}, e_{−α}, e_{−ν}, 1]

if ν < α, and by

G_{+} = [\begin{matrix}0 & − \frac{e_{ν}}{g_{−}} & 0 & − \frac{1}{f_{+}} & e_{λ+ν} & 0 \\ 0 & \frac{e_{ν−α}}{b_{+}g_{−}} & e_{ν} & 0 & − \frac{e_{2ν}}{b_{+}} & − \frac{d_{−}}{b_{+}} \\ − \frac{1}{d_{−}} & 0 & 0 & \frac{e_{ν}}{d_{−}f_{+}} & 0 & e_{λ} \\ 0 & 0 & b_{+} & 0 & 0 & 0 \\ 0 & − \frac{f_{+}}{g_{−}} & 0 & 0 & f_{+}e_{λ} & 0 \\ 0 & 0 & 0 & 0 & g_{−} & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{−}e_{−λ}}{f_{+}} & 1 \\ \frac{e_{−α}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 1 & \frac{e_{−α}}{d_{−}} & − \frac{e_{−2α}}{d_{−}f_{+}} & 0 \end{matrix} ]

(with the same Λ) if ν > α.

Case 11: d_{−} ≠ 0, f_{+} = 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}0 & − \frac{1}{g_{−}} & 0 & e_{λ+ν} & 0 & 0 \\ 0 & 0 & e_{ν} & − \frac{e_{2ν}}{b_{+}} & 0 & − \frac{d_{−}}{b_{+}} \\ − \frac{1}{d_{−}} & \frac{e_{ν}}{d_{−}g_{−}} & 0 & 0 & 0 & e_{λ} \\ 0 & 0 & b_{+} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & g_{−} & 0 & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \frac{e_{−α}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & \frac{e_{−α}}{d_{−}} & 0 & − \frac{e_{ν−α}}{d_{−}g_{−}} \end{matrix} ] ,

Λ = diag[e_{ν}, e_{ν}, e_{α}, e_{−ν}, e_{−λ}, 1]

if ν < α, and

G_{+} = [\begin{matrix}0 & −d_{−} & 0 & e_{λ+ν} & 0 & − \frac{e_{α}}{g_{−}} \\ 0 & 0 & e_{ν} & − \frac{e_{2ν}}{b_{+}} & 0 & \frac{1}{b_{+}g_{−}} \\ − \frac{1}{d_{−}} & e_{ν} & 0 & 0 & 0 & 0 \\ 0 & 0 & b_{+} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & g_{−} & 0 & 0 \end{matrix} ] ,

G_{−} = [ \begin{matrix}0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & \frac{e_{−α}}{d_{−}} & 0 & 0 \\ \frac{e_{−α}}{b_{+}} & 1 & \frac{d_{−}e_{−λ}}{b_{+}} & \frac{e_{−(λ+α)}}{b_{+}} & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & \frac{e_{−α}}{g_{−}} \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & −d_{−}g_{−}e_{α−ν} & −g_{−}e_{−ν} & 0 & 1 \end{matrix} ] ,

Λ = diag[e_{ν}, e_{α}, e_{α}, e_{−ν}, e_{−λ}, e_{ν−α}]

if ν > α.

Case 12: d_{−} = 0, f_{+} ≠ 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}e_{λ+ν} & 0 & 0 & − \frac{1}{f_{+}} & −e_{ν} & − \frac{e_{α}}{g_{−}} \\ 0 & e_{ν} & 0 & 0 & 0 & \frac{1}{b_{+}g_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ e_{ν} & b_{+} & 0 & 0 & 0 & 0 \\ f_{+}e_{λ} & 0 & 0 & 0 & −f_{+} & − \frac{f_{+}e_{α−ν}}{g_{−}} \\ g_{−} & 0 & 0 & 0 & 0 & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 & 0 & \frac{e_{−α}}{g_{−}} \\ 0 & 1 & 0 & \frac{e_{−(λ+α)}}{b_{+}} & 0 & − \frac{e_{−2α}}{b_{+}g_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & − \frac{e_{−α}}{f_{+}} & 0 \\ 0 & 0 & 0 & −g_{−}e_{−ν} & 0 & 1 \end{matrix} ] ,

Λ = diag[e_{−ν}, e_{α}, e_{λ}, e_{−α}, e_{−ν}, e_{ν−α}]

if ν < α, and

G_{+} = [\begin{matrix}e_{λ+ν} & 0 & 0 & − \frac{1}{f_{+}} & − \frac{e_{ν}}{g_{−}} & 0 \\ 0 & e_{ν} & 0 & 0 & \frac{e_{ν−α}}{b_{+}g_{−}} & − \frac{1}{b_{+}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ e_{ν} & b_{+} & 0 & 0 & 0 & 0 \\ f_{+}e_{λ} & 0 & 0 & 0 & − \frac{f_{+}}{g_{−}} & 0 \\ g_{−} & 0 & 0 & 0 & 0 & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & 0 & 0 & 0 & 0 & \frac{e_{−α}}{g_{−}} \\ 0 & 1 & 0 & \frac{e_{−(λ+α)}}{b_{+}} & 0 & − \frac{e_{−2α}}{b_{+}g_{−}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & − \frac{g_{−}e_{−λ}}{f_{+}} & 1 \\ 0 & 0 & 0 & 1 & − \frac{e_{−α}}{f_{+}} & 0 \end{matrix} ] ,

Λ = diag[e_{−ν}, e_{α}, e_{λ}, e_{−α}, 1, e_{−α}]

if ν > α.

Situation 6: Exactly one of the entries b_{±}, g_{±} differs from zero.

Using symmetries (2.8), (1), we may without loss of generality suppose that b_{−} ≠ 0, b_{+} = g_{±} = 0. Condition (4.5) and the first equation of (4.3) are satisfied automatically. The other two equations of (4.3) imply that l_{+} = f_{+} = 0. Applying elementary column operations, we can make a_{−} = d_{−} = 0 without changing any other entries. The further classification depends on l_{−}.

Case 13: l_{−} ≠ 0.

Using elementary column operations, we may assume that f_{−} = 0; these operations change the value of a_{+}. We now distinguish several subcases in terms of the new value of a_{+} and d_{+}.

Subcase 13a: a_{+}d_{+} ≠ 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}\frac{d_{+}}{a_{+}l_{−}} & 0 & −b_{−}e_{ν} & e_{α} & 0 & 0 \\ − \frac{d_{+}e_{ν}}{a_{+}b_{−}l_{−}} & − \frac{1}{b_{−}} & e_{2ν} & 0 & 0 & 0 \\ − \frac{1}{l_{−}} & 0 & 0 & − \frac{1}{d_{+}} & e_{λ+ν} & 0 \\ 0 & 0 & −a_{+}b_{−} & a_{+}e_{α−ν} & d_{+}e_{λ} & 0 \\ 0 & 0 & 0 & 0 & l_{−} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

G_{−} = [\begin{matrix}− \frac{l_{−}}{d_{+}} & − \frac{b_{−}l_{−}e_{−ν}}{d_{+}} & 0 & − \frac{l_{−}e_{−λ}}{d_{+}} & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ − \frac{e_{ν−α}}{a_{+}b_{−}} & 1 − \frac{e_{−α}}{a_{+}} & 0 & \frac{a_{+}e_{−α}−e_{−2α}}{a_{+}b_{−}} & \frac{d_{+}e_{ν−α}}{a_{+}b_{−}l_{−}} & 0 \\ 1 & b_{−}e_{−ν} & 0 & e_{−λ} & 0 & 0 \\ 0 & 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

Λ = diag[e_{ν}, e_{ν}, e_{α−ν}, e_{ν}, e_{−ν}, e_{−λ}]

if ν < α, and by

G_{+} = [ \begin{matrix}−b_{−} & 0 & e_{ν} & \frac{d_{+}e_{α}}{a_{+}l_{−}} & 0 & 0 \\ e_{ν} & − \frac{1}{b_{−}} & 0 & 0 & 0 & 0 \\ \frac{a_{+}b_{−}}{d_{+}} & 0 & − \frac{e_{ν−α}}{d_{+}} & − \frac{1+a_{+}e_{α}}{a_{+}l_{−}} & e_{λ+ν} & 0 \\ 0 & 0 & a_{+} & 0 & d_{+}e_{λ} & 0 \\ 0 & 0 & 0 & 0 & l_{−} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

G_{−} = [\begin{matrix}0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 + \frac{e_{−α}}{a_{+}} & \frac{b_{−}e_{−λ}}{a_{+}} & 0 & \frac{e_{−(λ+α)}}{a_{+}} & − \frac{d_{+}e_{−α}}{a_{+}l_{−}} & 0 \\ − \frac{l_{−}}{d_{+}} & \frac{b_{−}l_{−}(a_{+}e_{α−ν} −e_{−ν})}{d_{+}} & 0 & \frac{l_{−}(a_{+}e_{−ν} −e_{−λ})}{d_{+}} & 1 & 0 \\ 0 & 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

Λ = diag[e_{α}, e_{ν}, e_{α}, e_{ν−α}, e_{−ν}, e_{−λ}]

if ν > α.

Subcase 13b: a_{+} = 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}e_{λ} & 0 & \frac{d_{+}e_{α}}{l_{−}} & −b_{−} & 0 & 0 \\ 0 & − \frac{1}{b_{−}} & 0 & e_{ν} & 0 & 0 \\ 0 & 0 & − \frac{1}{l_{−}} & 0 & e_{λ+ν} & 0 \\ 1 & 0 & 0 & 0 & d_{+}e_{λ} & 0 \\ 0 & 0 & 0 & 0 & l_{−} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

G_{−} = [\begin{matrix}1 & b_{−}e_{−ν} & 0 & e_{−λ} & − \frac{d_{+}}{l_{−}} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & 0 & 0 \\ 0 & 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

Λ = diag[1, e_{ν}, e_{ν}, e_{α}, e_{−ν}, e_{−λ}].

Subcase 13c: a_{+} ≠ 0, d_{+} = 0. Then G can be represented as G = X_{+}G′X_{−}, where

X_{+} = [\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ − \frac{a_{+}e_{λ}+e_{ν}}{b_{−}} & − \frac{1}{b_{−}} & 0 & e_{λ+ν} & 0 & 0 \\ 0 & 0 & − \frac{1}{l_{−}} & 0 & e_{λ+ν} & 0 \\ 0 & 0 & 0 & b_{−} & 0 & 0 \\ 0 & 0 & 0 & 0 & l_{−} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]

is invertible in AP_{W}^{+} ,

X_{−} = [\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & 0 & 0 \\ 0 & 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]

is invertible in AP_{W}^{−}, and

G′ = [ \begin{matrix}e_{λ} & 0 & 0 & 0 & 0 & 0 \\ 0 & e_{ν} & 0 & 0 & 0 & 0 \\ 0 & 0 & e_{ν} & 0 & 0 & 0 \\ \frac{1+a_{+}e_{α}}{b_{−}} & 0 & 0 & e_{−ν} & 0 & 0 \\ 0 & 0 & 0 & 0 & e_{−ν} & 0 \\ 0 & 0 & 0 & 0 & 0 & e_{−λ} \end{matrix}] .

Obviously, G′ can be represented as a direct sum of a diagonal matrix diag[e_{ν}, e_{ν}, e_{−ν}, e_{−λ}] with a 2 × 2 matrix

e_{α/2} [ \begin{matrix}e_{ν+α/2} & 0 \\ \frac{e_{−α/2}+a_{+}e_{α/2}}{b_{−}} & e_{−ν−α/2} \end{matrix}] .

The latter matrix is AP_{W} factorable due to Theorem 1.2. This in turn guarantees the AP_{W} factorability of G.

Case 14: l_{−} = 0.

If in addition d_{+} = 0, then G splits into a direct sum of a diagonal matrix diag[e_{λ}, e_{−λ}] with another matrix G_{1} of the type (1.4), for which m = 2 and

c_{−1} = [\begin{matrix}b_{−} & 0 \\ 0 & f_{−} \end{matrix}] , c_{1} = [\begin{matrix}0 & a_{+} \\ 0 & 0 \end{matrix}] , c_{0} = [\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}] .

The matrix G_{1} satisfies Corollary 2.5 if f_{−} ≠ 0, and falls into Case 3 of Appendix B otherwise. Either way, G_{1} is AP_{W} factorable. The AP_{W} factorability of G follows immediately.

If d_{+} ≠ 0, we can use an elementary column transformation to make a_{+} = 0. It remains to consider two subcases, depending on the value of f_{−}.

Subcase 14a: f_{−} ≠ 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}0 & 0 & 0 & − \frac{1}{f_{−}} & e_{λ+ν} & 0 \\ 0 & − \frac{1}{b_{−}} & e_{λ} & \frac{e_{ν}}{b_{−}f_{−}} & 0 & 0 \\ e_{ν} & 0 & − \frac{b_{−}}{d_{+}} & 0 & 0 & 0 \\ d_{+} & 0 & 0 & 0 & e_{ν} & 0 \\ 0 & 0 & 0 & 0 & f_{−} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

G_{−} = [\begin{matrix}0 & \frac{b_{−}e_{−λ}}{d_{+}} & 1 & \frac{e_{−(λ+α)}}{d_{+}} & − \frac{e_{−2α}}{d_{+}f_{−}} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & − \frac{e_{ν−α}}{b_{−}f_{−}} & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & \frac{e_{−α}}{f_{−}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

Λ = diag[e_{α}, e_{ν}, 1, e_{ν}, e_{−ν}, e_{−λ}]

if ν < α, and by

G_{+} = [\begin{matrix}0 & 0 & − \frac{e_{α}}{f_{−}} & −b_{−} & e_{λ+ν} & 0 \\ 0 & − \frac{1}{b_{−}} & 0 & e_{ν} & 0 & 0 \\ e_{ν} & 0 & \frac{1}{d_{+}f_{−}} & 0 & 0 & 0 \\ d_{+} & 0 & 0 & 0 & e_{ν} & 0 \\ 0 & 0 & 0 & 0 & f_{−} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

G_{−} = [\begin{matrix}0 & \frac{b_{−}e_{−λ}}{d_{+}} & 1 & \frac{e_{−(λ+α)}}{d_{+}} & − \frac{e_{−2α}}{d_{+}f_{−}} & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & −b_{−}f_{−}e_{α−ν} & 0 & −f_{−}e_{−ν} & 1 & 0 \\ 0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & 0 & 0 \\ 1 & 0 & 0 & 0 & \frac{e_{−α}}{f_{−}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

Λ = diag[e_{α}, e_{ν}, e_{ν−α}, e_{α}, e_{−ν}, e_{−λ}]

if ν > α .

Subcase 14b: f_{−} = 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}0 & 0 & e_{α} & −b_{−} & 0 & 0 \\ 0 & − \frac{1}{b_{−}} & 0 & e_{ν} & 0 & 0 \\ e_{ν} & 0 & − \frac{1}{d_{+}} & 0 & 0 & 0 \\ d_{+} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

G_{−} = [ \begin{matrix}\frac{e_{−α}}{d_{+}} & \frac{b_{−}e_{−λ}}{d_{+}} & 1 & \frac{e_{−(λ+α)}}{d_{+}} & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & b_{−}e_{−ν} & 0 & e_{−λ} & 0 & 0 \\ 0 & 1 & 0 & \frac{e_{−α}}{b_{−}} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] ,

Λ = diag[e_{α}, e_{ν}, e_{ν}, e_{α}, e_{−λ}, e_{−λ}].

Appendix D.

c_{−1} = [\begin{matrix}a_{−} & 0 \\ 0 & l_{−} \end{matrix}] , c_{0} = [\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}] , c_{1} = [\begin{matrix}a_{+} & d_{+} \\ f_{+} & l_{+} \end{matrix}] ,

where l_{−} ≠ 0 and either l_{+} = 0 and exactly one of d_{+}, f_{+} differs from zero (Situation 1), or l_{+} ≠ 0, a_{−} det c_{1} = 0 (Situation 2).

In Situation 1, we can use symmetry (1) to suppose without loss of generality that d_{+} = 0, f_{+} ≠ 0. We now consider the four separate cases, depending on the zero/non-zero pattern of a_{±}.

Case 1: a_{±} ≠ 0.

For ν < α, G can be represented as G = X_{+}G′X_{−}, where

X_{+} = [\begin{matrix}e_{λ} − \frac{a_{−}}{a_{+}} & \frac{a_{+}l_{−}e_{ν}}{f_{+}} & − \frac{l_{−}}{f_{+}} & 0 \\ \frac{a_{−}f_{+}e_{λ}}{a_{+}l_{−}} & 0 & e_{λ} & − \frac{1}{l_{−}} \\ a_{+}e_{α} + 1 & \frac{a_{+}^{2}l_{−}}{f_{+}} & 0 & 0 \\ f_{+}e_{α} & a_{+}l_{−} & 0 & 0 \end{matrix} ]

is invertible in AP_{W}^{+} ,

X_{−} = [\begin{matrix}1 + a_{−}e_{−ν} & − \frac{a_{+}l_{−}e_{−ν}}{f_{+}} & e_{−λ} & − \frac{a_{+}e_{−λ}}{f_{+}} \\ − \frac{a_{−}f_{+}}{a_{+}l_{−}} & 1 + \frac{e_{−α}}{a_{+}} & − \frac{f_{+}e_{−α}}{a_{+}l_{−}} & \frac{e_{−2α}+a_{+}e_{−α}}{a_{+}l_{−}} \\ 0 & − \frac{1+a_{−}e_{−ν}}{a_{+}} & \frac{f_{+}}{a_{+}l_{−}} & − \frac{a_{−}e_{−λ}+e_{−α}}{a_{+}l_{−}} \\ 0 & 0 & 0 & 1 \end{matrix} ]

is invertible in AP_{W}^{−}, and

G′ = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & e_{α−ν} & 0 & 0 \\ 0 & a_{−}e_{−ν} + 1 & e_{−α} & 0 \\ 0 & 0 & 0 & e_{ν} \end{matrix} ]

splits into a direct sum of a diagonal matrix diag[1, e_{ν}] with the matrix

e_{−ν/2} [\begin{matrix}e_{α−ν/2} & 0 \\ a_{−}e_{−ν/2} + e_{ν/2} & e_{−α+ν/2} \end{matrix}] .

The latter matrix is AP_{W} factorable due to Theorem 1.2. This implies the AP_{W} factorability of G′ and ultimately of G itself.

When ν > α, the AP_{W} factorization of G is delivered by

G_{+} = [\begin{matrix}− \frac{e_{λ}}{a_{−}} & \frac{l_{−}(a_{+}e_{ν} −e_{ν−α})}{f_{+}} & e_{λ} − \frac{a_{−}}{a_{+}} & 0 \\ 0 & e_{2ν} & \frac{a_{−}f_{+}e_{λ}}{a_{+}l_{−}} & − \frac{1}{l_{−}} \\ − \frac{1+a_{+}e_{α}}{a_{−}} & \frac{a_{+}^{2}l_{−}}{f_{+}} & 1 + a_{+}e_{α} & 0 \\ − \frac{f_{+}e_{α}}{a_{−}} & a_{+}l_{−} & f_{+}e_{α} & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}−a_{−}^{2}e_{α−ν} & \frac{l_{−}(a_{+}a_{−}e_{α−ν} −1)}{f_{+}} & 1 − a_{−}e_{−ν} & \frac{a_{+}a_{−}e_{−ν} −e_{−α}}{f_{+}} \\ − \frac{a_{−}f_{+}}{a_{+}l_{−}} & 1 + \frac{e_{−α}}{a_{+}} & − \frac{f_{+}e_{−α}}{a_{+}l_{−}} & \frac{e_{−2α}+a_{+}e_{−α}}{a_{+}l_{−}} \\ 1 & − \frac{l_{−}e_{−α}}{a_{−}f_{+}} & \frac{e_{−α}}{a_{−}} & − \frac{e_{−2α}}{a_{−}f_{+}} \\ 0 & 0 & 0 & 1 \end{matrix} ] ,

Λ = diag[e_{−α}, e_{α−ν}, 1, e_{ν}].

Case 2: a_{+} ≠ 0, a_{−} = 0. The AP_{W} factorization of G is delivered by

G_{+} = [\begin{matrix}e_{λ} & \frac{l_{−}e_{ν}}{f_{+}} & 1 − a_{+}e_{α} & 0 \\ 0 & 0 & − \frac{f_{+}e_{λ}}{l_{−}} & − \frac{1}{l_{−}} \\ 1 + a_{+}e_{α} & \frac{a_{+}l_{−}}{f_{+}} & −a_{+}^{2}e_{α−ν} & 0 \\ f_{+}e_{α} & l_{−} & −a_{+}f_{+}e_{α−ν} & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}1 & − \frac{a_{+}l_{−}e_{−ν}}{f_{+}} & e_{−λ} & − \frac{a_{+}e_{−λ}}{f_{+}} \\ 0 & 1 & − \frac{f_{+}}{l_{−}} & \frac{e_{−α}}{l_{−}} \\ 0 & − \frac{l_{−}}{f_{+}} & 0 & − \frac{e_{−α}}{f_{+}} \\ 0 & 0 & 0 & 1 \end{matrix} ] ,

Λ = diag[1, e_{−ν}, 1, e_{ν}]

when ν < α, and by

G_{+} = [\begin{matrix}1 & \frac{l_{−}(a_{+}e_{ν} −e_{ν−α})}{a_{+}f_{+}} & e_{λ} & 0 \\ − \frac{f_{+}e_{λ}}{l_{−}} & \frac{e_{2ν}}{a_{+}} & 0 & − \frac{1}{l_{−}} \\ 0 & \frac{a_{+}l_{−}}{f_{+}} & 1 + a_{+}e_{α} & 0 \\ 0 & l_{−} & f_{+}e_{α} & 0 \end{matrix} ] ,

G_{−} = [\begin{matrix}0 & \frac{l_{−}}{a_{+}f_{+}} & − \frac{1}{a_{+}} & \frac{e_{−α}}{a_{+}f_{+}} \\ 0 & e_{−α} + a_{+} & − \frac{f_{+}e_{−α}}{l_{−}} & \frac{e_{−2α}+a_{+}e_{−α}}{l_{−}} \\ 1 & − \frac{a_{+}l_{−}e_{−ν}}{f_{+}} & e_{−λ} & − \frac{a_{+}e_{−λ}}{f_{+}} \\ 0 & 0 & 0 & 1 \end{matrix} ] ,

Λ = diag[e_{−α}, e_{α−ν}, 1, e_{ν}]

if ν > α.

Case 3: a_{+} = 0, a_{−} ≠ 0. If ν < α, G can be represented as G = X_{+}G′X_{−}, where

X_{+} = [\begin{matrix}− \frac{l_{−}}{f_{+}} & 0 & −a_{−}e_{ν} & e_{λ} \\ e_{λ} & − \frac{1}{l_{−}} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −a_{−}f_{+} & f_{+}e_{α} \end{matrix}]

is invertible in AP_{W}^{+} ,

X_{−} = [\begin{matrix}0 & 1 & 0 & \frac{e_{−α}}{l_{−}} \\ 0 & 0 & 0 & 1 \\ 1 & − \frac{l_{−}e_{−α}}{a_{−}f_{+}} & \frac{e_{−α}}{a_{−}} & − \frac{e_{−2α}}{a_{−}f_{+}} \\ 0 & \frac{l_{−}(1+a_{−}e_{−ν})}{a_{−}f_{+}} & − \frac{1}{a_{−}} & \frac{a_{−}e_{−λ}+e_{−α}}{a_{−}f_{+}} \end{matrix} ]

is invertible in AP_{W}^{−}, and

G′ = [\begin{matrix}1 & 0 & 0 & 0 \\ 0 & e_{ν} & 0 & 0 \\ 0 & 0 & e_{α−ν} & 0 \\ 0 & 0 & 1 + a_{−}e_{−ν} & e_{−α} \end{matrix}] .

As in Case 1, G′ splits into a direct sum of a diagonal matrix and another matrix of the type (1.4), AP_{W} factorable due to Theorem 1.2. This implies AP_{W} factorability of G.

If ν > α, an explicit AP_{W} factorization of G is delivered by

G_{+} = [\begin{matrix}e_{2ν} − a_{−}e_{ν} & 0 & − \frac{l_{−}}{f_{+}} & e_{λ} \\ 0 & − \frac{1}{l_{−}} & e_{λ} & 0 \\ e_{ν−α} & 0 & 0 & 1 \\ f_{+}(e_{ν} − a_{−}) & 0 & 0 & f_{+}e_{α} \end{matrix}] ,

G_{−} = [\begin{matrix}1 & − \frac{l_{−}e_{−α}}{a_{−}f_{+}} & \frac{e_{−α}}{a_{−}} & − \frac{e_{−2α}}{a_{−}f_{+}} \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} \\ a_{−}e_{α−ν} & \frac{l_{−}}{a_{−}f_{+}} & e_{−ν} − \frac{1}{a_{−}} & \frac{e_{−α}}{a_{−}f_{+}} \end{matrix} ] ,

Λ = diag[e_{α−ν}, e_{ν}, 1, e_{−α}].

Case 4: a_{+} = a_{−} = 0. The AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}− \frac{l_{−}}{f_{+}} & 0 & e_{λ} & −e_{ν} \\ e_{λ} & − \frac{1}{l_{−}} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & f_{+}e_{α} & −f_{+} \end{matrix}] , G_{−} = [\begin{matrix}0 & 1 & 0 & \frac{e_{−α}}{l_{−}} \\ 0 & 0 & 0 & 1 \\ 1 & 0 & e_{−λ} & 0 \\ 0 & − \frac{l_{−}}{f_{+}} & 1 & − \frac{e_{−α}}{f_{+}} \end{matrix} ] ,

Λ = diag[1, e_{ν}, 1, e_{−ν}].

In Situation 2, we distinguish between the following cases, depending on D = det c_{1}. Case 5: D ≠ 0. Then a_{−} = 0, and G can be represented as G = X_{+}G′X_{−}, where

X_{+} = [\begin{matrix}1 & \frac{d_{+}l_{−}e_{α}}{l_{+}} & 0 & 0 \\ − \frac{f_{+}}{l_{+}} & e_{λ} − \frac{l_{−}}{l_{+}} & − \frac{e_{ν}}{l_{−}} & \frac{f_{+}e_{ν}}{l_{+}D} \\ 0 & \frac{d_{+}(l_{+}e_{α}+a_{+}l_{−}e_{α−ν})}{l_{+}} & − \frac{d_{+}}{l_{−}} & \frac{a_{+}}{D} \\ 0 & l_{+}e_{α} + \frac{d_{+}f_{+}l_{−}e_{α−ν}}{l_{+}} & − \frac{l_{+}}{l_{−}} & \frac{f_{+}}{D} \end{matrix} ] ,

X_{−} = [ \begin{matrix}1 & − \frac{d_{+}l_{−}e_{−ν}}{l_{+}} & 0 & − \frac{d_{+}e_{−λ}}{l_{+}} \\ 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} \\ \frac{f_{+}l_{−}}{D} & − \frac{a_{+}l_{−}^{2}e_{−ν}}{D} & \frac{f_{+}l_{−}e_{−λ}}{D} & 1 − \frac{a_{+}l_{−}e_{−λ}}{D} \\ 0 & 0 & l_{+} & 0 \end{matrix} ] , G′ = [\begin{matrix}e_{λ} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ l_{+} + De_{α} & 0 & 0 & e_{−λ} \end{matrix}]

if ν < α, and

X_{+} = [\begin{matrix}\frac{l_{+}^{2}}{D^{2}} & \frac{d_{+}l_{−}}{D} & 0 & \frac{e_{ν}}{D} − \frac{l_{+}e_{ν−α}}{D^{2}} \\ − \frac{f_{+}l_{+}}{D^{2}} & e_{λ} − \frac{a_{+}l_{−}}{D} & − \frac{e_{ν}}{l_{−}} & \frac{f_{+}e_{ν−α}}{D^{2}} \\ 0 & d_{+}e_{α} & − \frac{d_{+}}{l_{−}} & \frac{a_{+}}{D} \\ 0 & l_{+}e_{α} & − \frac{l_{+}}{l_{−}} & \frac{f_{+}}{D} \end{matrix} ] , G′ = [\begin{matrix}e_{ν−α} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ l_{+} + De_{α} & 0 & 0 & e_{α−ν} \end{matrix} ] ,

X_{−} = [ \begin{matrix}1 & − \frac{d_{+}l_{−}e_{−ν}}{l_{+}} & e_{−λ} − a_{+}e_{−ν} + \frac{d_{+}f_{+}e_{−ν}}{l_{+}} & − \frac{d_{+}e_{−λ}}{l_{+}} \\ 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} \\ \frac{f_{+}l_{−}}{D} & − \frac{a_{+}l_{−}^{2}e_{−ν}}{D} & \frac{f_{+}l_{−}e_{−λ}}{D} & 1 − \frac{a_{+}l_{−}e_{−λ}}{D} \\ 0 & \frac{d_{+}l_{−}D}{l_{+}} & \frac{D^{2}}{l_{+}} & \frac{d_{+}De_{−α}}{l_{+}} \end{matrix} ]

if ν > α.

In both cases, X_{+} is invertible in AP_{W}^{+} , X_{−} is invertible in AP_{W}^{−}, and G′ splits into a direct sum of I_{2} with yet another 2 × 2 matrix G_{2} of type (1.4), AP_{W} factorable due to Theorem 1.2. From AP_{W} factorability of G_{2} we conclude that G′ is AP_{W} factorable as well. This in turn implies the AP_{W} factorability of G.

Case 6: D=0.

Applying an elementary row operation, we may without loss of generality suppose that a_{+} = d_{+} = 0. However, this changes the upper right entry of c_{−1} so that d_{−} is now not necessarily zero. In other words, we need to consider matrices (1.4), (1.5) with

c_{−1} = [\begin{matrix}a_{−} & d_{−} \\ 0 & l_{−} \end{matrix}] , c_{0} = [\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}] , c_{1} = [\begin{matrix}0 & 0 \\ f_{+} & l_{+} \end{matrix}]

and l_{±} ≠ 0.

We partition this case into three subcases.

Subcase 6a: a_{−}d_{−} ≠ 0.

If ν < α, G can be represented as G = X_{+}G′X_{−}, where

X_{+} = [\begin{matrix}a_{−}d_{−} & 0 & −d_{−}e_{α} & \frac{e_{λ}}{l_{+}} \\ − \frac{d_{−}f_{+}(a_{−}+e_{ν})}{l_{+}} & − \frac{e_{ν}}{l_{−}} & e_{λ} + \frac{d_{−}f_{+}e_{α}−l_{−}}{l_{+}} & − \frac{a_{−}(l_{+}e_{λ}−l_{−})}{d_{−}l_{+}^{2}} \\ 0 & 0 & 0 & \frac{1}{l_{+}} \\ −d_{−}f_{+} & − \frac{l_{+}}{l_{−}} & l_{+}e_{α} & \frac{(d_{−}f_{+}−a_{−}l_{+})e_{α}}{d_{−}l_{+}} \end{matrix} ]

is invertible in AP_{W}^{+} ,

X_{−} = [\begin{matrix}\frac{a_{−}}{d_{−}} & 1 & \frac{a_{−}e_{−α}−e_{ν−α}}{a_{−}d_{−}} & \frac{e_{ν−α}}{a_{−}l_{−}} \\ 0 & − \frac{l_{−}^{2}e_{−ν}}{l_{+}} & \frac{f_{+}l_{−}}{a_{−}l_{+}} & 1 − \frac{a_{−}l_{−}e_{−λ}+d_{−}f_{+}}{a_{−}l_{+}} \\ 0 & 0 & \frac{1}{a_{−}d_{−}} & − \frac{1}{a_{−}l_{−}} \\ l_{+}(a_{−}e_{−ν} + 1) & d_{−}l_{+}e_{−ν} & l_{+}e_{−λ} & 0 \end{matrix} ]

is invertible in AP_{W}^{−}, and

G′ = [\begin{matrix}e_{α−ν} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ a_{−}e_{−ν} + 1 & 0 & e_{ν−α} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

splits into a direct sum of I_{2} and a 2 × 2 matrix [\begin{matrix}e_{α−ν} & 0 \\ a_{−}e_{−ν} + 1 & e_{ν−α} \end{matrix}], AP_{W} factorable due

to Theorem 1.2.

When ν > α the AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}−d_{−}e_{α} & 0 & −a_{−}d_{−} & \frac{e_{λ}}{l_{+}} \\ e_{λ} + \frac{d_{−}f_{+}e_{α}−l_{−}}{l_{+}} & − \frac{e_{ν}}{l_{−}} & \frac{a_{−}d_{−}f_{+}}{l_{+}} & − \frac{a_{−}(l_{+}e_{λ}−l_{−})}{d_{−}l_{+}^{2}} \\ 0 & 0 & 0 & \frac{1}{l_{+}} \\ l_{+}e_{α} & − \frac{l_{+}}{l_{−}} & 0 & \frac{(d_{−}f_{+}−a_{−}l_{+})e_{α}}{d_{−}l_{+}} \end{matrix} ] ,

G_{−} = [\begin{matrix}\frac{a_{−}(a_{−}e_{−ν} +1)}{d_{−}} & a_{−}e_{−ν} + 1 & \frac{a_{−}e_{−λ}}{d_{−}} & \frac{e_{−α}}{l_{−}} \\ \frac{a_{−}f_{+}l_{−}e_{α−ν}}{l_{+}} & \frac{l_{−}(d_{−}f_{+}e_{α−ν} −l_{−}e_{−ν})}{l_{+}} & \frac{l_{−}f_{+}e_{−ν}}{l_{+}} & 1 − \frac{l_{−}e_{−λ}}{l_{+}} \\ − \frac{a_{−}e_{α−ν}}{d_{−}} & −e_{α−ν} & \frac{1−a_{−}e_{−ν}}{a_{−}d_{−}} & − \frac{1}{a_{−}l_{−}} \\ l_{+}(1 + a_{−}e_{−ν}) & d_{−}l_{+}e_{−ν} & l_{+}e_{−λ} & 0 \end{matrix} ]

and its partial AP indices equal zero, that is, Λ = I. Subcase 6b: a_{−} = 0. The AP_{W} factorization of G is delivered by

G_{+} = [\begin{matrix}−d_{−}e_{α} & 0 & −1 & \frac{e_{λ}}{l_{+}} \\ e_{λ} − \frac{l_{−}}{l_{+}} & − \frac{e_{ν}}{l_{−}} & \frac{f_{+}}{l_{+}} & 0 \\ 0 & 0 & 0 & \frac{1}{l_{+}} \\ l_{+}e_{α} − d_{−}f_{+}e_{α−ν} & − \frac{l_{+}}{l_{−}} & 0 & \frac{f_{+}e_{α}}{l_{+}} \end{matrix} ] ,

G_{−} = [\begin{matrix}0 & 1 & 0 & \frac{e_{−α}}{l_{−}} \\ 0 & − \frac{l_{−}^{2}e_{−ν}}{l_{+}} & \frac{f_{+}l_{−}e_{−ν}}{l_{+}} & 1 − \frac{l_{−}e_{−λ}+d_{−}f_{+}e_{−ν}}{l_{+}} \\ 0 & 0 & 1 & − \frac{d_{−}}{l_{−}} \\ l_{+} & d_{−}l_{+}e_{−ν} & l_{+}e_{−λ} & 0 \end{matrix} ]

if ν < α, and

G_{+} = [\begin{matrix}−d_{−}e_{α} & 0 & −1 & \frac{e_{λ}}{l_{+}} \\ e_{λ} − \frac{d_{−}f_{+}e_{α}−l_{−}}{l_{+}} & − \frac{e_{ν}}{l_{−}} & \frac{f_{+}}{l_{+}} & 0 \\ 0 & 0 & 0 & \frac{1}{l_{+}} \\ l_{+}e_{α} & − \frac{l_{+}}{l_{−}} & 0 & \frac{f_{+}e_{α}}{l_{+}} \end{matrix} ] ,

G_{−} = [\begin{matrix}0 & 1 & 0 & \frac{e_{−α}}{l_{−}} \\ 0 & \frac{l_{−}(d_{−}f_{+}e_{α−ν} −l_{−}e_{−ν})}{l_{+}} & \frac{f_{+}l_{−}e_{−ν}}{l_{+}} & 1 − \frac{l_{−}e_{−λ}}{l_{+}} \\ 0 & 0 & 1 & − \frac{d_{−}}{l_{−}} \\ l_{+} & d_{−}l_{+}e_{−ν} & l_{+}e_{−λ} & 0 \end{matrix} ]

if ν > α; in both cases all partial AP indices of G equal zero. Subcase 6c: a_{−} ≠ 0, d_{−} = 0. If ν < α, G can be represented as G = X_{+}G′X_{−}, where

X_{+} = [ \begin{matrix}−a_{−} & 0 & e_{λ} & 0 \\ \frac{f_{+}(a_{−}+e_{ν})}{l_{+}} & e_{λ} − \frac{l_{−}}{l_{+}} & − \frac{f_{+}e_{λ}}{l_{+}} & − \frac{e_{ν}}{l_{−}} \\ 0 & 0 & 1 & 0 \\ f_{+} & l_{+}e_{α} & 0 & − \frac{l_{+}}{l_{−}} \end{matrix} ]

is invertible in AP_{W}^{+} ,

X_{−} = [\begin{matrix}1 & 0 & \frac{e_{−α}}{a_{−}} & 0 \\ 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} \\ 0 & 0 & − \frac{1}{a_{−}} & 0 \\ 0 & − \frac{l_{−}^{2}e_{−ν}}{l_{+}} & \frac{f_{+}l_{−}}{a_{−}l_{+}} & 1 − \frac{l_{−}e_{−λ}}{l_{+}} \end{matrix} ]

is invertible in AP_{W}^{−}, and

G′ = [\begin{matrix}e_{α} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 1 + a_{−}e_{−ν} & 0 & e_{−α} & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

splits into a direct sum of I_{2} and the matrix [\begin{matrix}e_{α} & 0 \\ 1 + a_{−}e_{−ν} & e_{−α} \end{matrix}], AP_{W} factorable due to

Theorem 1.2.

When ν > α, G is AP_{W} factorable with zero partial AP indices and

G_{+} = [\begin{matrix}a_{−} & 0 & e_{λ} − a_{−}e_{α} & 0 \\ − \frac{f_{+}(a_{−}+e_{ν})}{l_{+}} & e_{λ} − \frac{l_{−}}{l_{+}} & \frac{a_{−}f_{+}e_{α}}{l_{+}} & − \frac{e_{ν}}{l_{−}} \\ 0 & 0 & 1 & 0 \\ −f_{+} & l_{+}e_{α} & f_{+}e_{α} & − \frac{l_{+}}{l_{−}} \end{matrix} ] ,

G_{−} = [\begin{matrix}a_{−}e_{α−ν} & 0 & e_{−ν} − \frac{1}{a_{−}} & 0 \\ 0 & 1 & 0 & \frac{e_{−α}}{l_{−}} \\ 1 + a_{−}e_{−ν} & 0 & e_{−λ} & 0 \\ 0 & − \frac{l_{−}^{2}e_{−ν}}{l_{+}} & \frac{f_{+}l_{−}}{a_{−}l_{+}} & 1 − \frac{l_{−}e_{−λ}}{l_{+}} \end{matrix} ] .

Appendix E.

c_{±1} = [\begin{matrix}d_{±} & f_{±} & b_{±} \\ 0 & d_{±} & 0 \\ 0 & a_{±} & 0 \end{matrix}] , c_{0} = [\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}] ,

where

a_{+}a_{−}b_{+}b_{−} ≠ 0 and a_{+}b_{−} = a_{−}b_{+}.

In this appendix we shall use the following abbreviations:

A = a_{−}d_{+} − a_{+}d_{−}  B = a_{−}f_{+} − a_{+}f_{−}
C = b_{−}d_{+} − b_{+}d_{−}  D = b_{−}f_{+} − b_{+}f_{−}.

Case 1: C ≠ 0.

The AP_{W} factorization of G is given by

G_{+} = ⌈\begin{matrix}−b_{−}e_{ν} & 0 & e_{α} \\ 0 & e_{λ+ν} − \frac{a_{−}}{a_{+}} e_{ν} & 0 \\ e_{2ν} + d_{−}e_{ν} & \frac{a_{−}f_{+}e_{ν}}{a_{+}b_{+}} − \frac{f_{−}e_{λ+ν}}{b_{−}} & − \frac{d_{−}e_{α}}{b_{−}} − \frac{1}{b_{+}} \\ b_{+}e_{ν} + b_{+}d_{−} − d_{+}b_{−} & \frac{De_{λ}}{b_{−}} & \frac{Be_{α−ν}}{b_{−}} \\ 0 & e_{ν} + d_{−} + d_{+}e_{λ} − \frac{a_{−}d_{+}}{a_{+}} & 0 \\ 0 & a_{+}e_{λ} & 0 \end{matrix}

\begin{matrix}\frac{b_{+}}{C} & 0 & 0 \\ 0 & − \frac{e_{α}}{a_{−}} − \frac{a_{+}e_{λ}−a_{−}}{a_{−}A} & −e_{ν} \\ − \frac{b_{+}e_{ν} +d_{+}b_{−}}{b_{−}C} & \frac{f_{+}e_{α}}{a_{−}b_{+}} − \frac{a_{−}b_{−}f_{+}−a_{+}b_{+}f_{−}e_{λ}}{a_{−}b_{−}b_{+}A} & \frac{f_{−}e_{ν}}{b_{+}} \\ − \frac{b_{+}^{2}}{b_{−}C} & − \frac{a_{+}De_{α}}{a_{−}b_{−}A} & 0 \\ 0 & \frac{d_{+}e_{α−ν}}{a_{−}} − \frac{a_{+}+a_{+}d_{+}e_{α}}{a_{−}A} & −d_{+} \\ 0 & − \frac{a_{+}e_{α−ν}}{a_{−}} − \frac{a_{+}^{2}e_{α}}{a_{−}A} & −a_{+} \end{matrix} ⌋ ,

G_{−} = ⌈ \begin{matrix}\frac{d_{−}}{b_{−}} − \frac{d_{−}e_{−α}+e_{ν−α}}{C} & \frac{f_{−}}{b_{−}} − \frac{f_{−}e_{−α}}{C} & 1 − \frac{b_{−}e_{−α}}{C} \\ 0 & 1 & 0 \\ 1 + d_{−}e_{−ν} & f_{−}e_{−ν} & b_{−}e_{−ν} \\ − \frac{b_{−}(1+d_{−}e_{−ν})}{b_{+}} & − \frac{b_{−}f_{−}e_{−ν}}{b_{+}} & − \frac{b_{−}^{2}e_{−ν}}{b_{+}} \\ 0 & 0 & 0 \\ 0 & − \frac{a_{−}}{a_{+}} & 0 \end{matrix}

\begin{matrix}\frac{e_{−α}}{b_{−}} − \frac{b_{−}e_{−2α}−b_{+}e_{ν−α}}{C} & 0 & \frac{De_{ν−α}}{a_{−}b_{−}C} \\ 0 & − \frac{a_{+}e_{−α}}{A} & \frac{a_{+}e_{ν−α}+a_{−}d_{+}e_{−α}}{a_{−}A} \\ e_{−λ} & 0 & 0 \\ 1 − \frac{b_{−}e_{−λ}}{b_{+}} & 0 & \frac{D}{a_{−}b_{+}} \\ 0 & −a_{−}e_{−ν} & 1 + d_{−}e_{−ν} \\ 0 & 1 & − \frac{a_{+}d_{−}+a_{−}e_{−α}}{a_{+}a_{−}} \end{matrix} ⌋ ,

Λ = diag[e_{α−ν}, e_{−ν}, e_{−ν}, e_{ν}, e_{ν}, e_{ν}, e_{ν−α}, e_{−ν}]

if ν < α, and by

G_{+} = ⌈\begin{matrix}−b_{−} & 0 & \frac{b_{+}e_{α}}{C} \\ 0 & \frac{a_{+}e_{λ}−a_{−}}{A} & 0 \\ e_{ν} + \frac{b_{−}d_{+}}{b_{+}} & \frac{a_{−}b_{−}f_{+}−a_{+}f_{−}b_{+}e_{λ}}{b_{−}b_{+}A} & − \frac{1+d_{+}e_{α}}{C} \\ b_{+} & \frac{Da_{+}e_{α}}{Ab_{−}} & 0 \\ 0 & \frac{a_{+}(1+d_{+}e_{α})}{A} & 0 \\ 0 & \frac{a_{+}^{2}e_{α}}{A} & 0 \end{matrix}

\begin{matrix}e_{ν} & 0 & 0 \\ 0 & − \frac{e_{λ}}{a_{−}} & e_{2ν} − d_{−}e_{ν} + \frac{a_{−}d_{+}e_{ν} −a_{−}e_{ν−α}}{a_{+}} \\ − \frac{e_{ν−α}}{b_{+}} − \frac{d_{−}e_{ν}}{b_{−}} & \frac{f_{+}e_{λ}}{a_{−}b_{+}} & \frac{a_{−}b_{−}f_{+}e_{ν−α}−b_{−}f_{+}Ae_{ν} −a_{+}b_{+}f_{−}e_{2ν}}{a_{+}b_{−}b_{+}} \\ \frac{C}{b_{−}} & 0 & \frac{De_{ν}}{b_{−}} \\ 0 & − \frac{1+d_{+}e_{α}}{a_{−}} & d_{+}e_{ν} − d_{−}d_{+} + e_{ν−α} + \frac{a_{−}d_{+}^{2}}{a_{+}} \\ 0 & − \frac{a_{+}e_{α}}{a_{−}} & A + a_{+}e_{ν} \end{matrix} ⌋ ,

G_{−} = ⌈ \begin{matrix}\frac{d_{−}}{b_{−}} & \frac{f_{−}}{b_{−}} & 1 \\ 0 & − \frac{a_{−}}{a_{+}} & 0 \\ \frac{d_{−}Ce_{α−ν} −b_{−}−b_{−}d_{−}e_{−ν}}{b_{+}} & \frac{f_{−}(Ce_{α−ν} −b_{−}e_{−ν})}{b_{+}} & \frac{b_{−}(Ce_{α−ν} −b_{−}e_{−ν})}{b_{+}} \\ 1 + b_{−} \frac{e_{−α}+d_{−}e_{−λ}}{C} & \frac{b_{−}f_{−}e_{−λ}}{C} & \frac{b_{−}^{2}e_{−λ}}{C} \\ 0 & \frac{a_{−}Ae_{α−ν}}{a_{+}} & 0 \\ 0 & 1 + \frac{a_{−}e_{−α}}{A} & 0 \end{matrix}

\begin{matrix}\frac{e_{−α}}{b_{−}} & 0 & 0 \\ 0 & 1 & − \frac{d_{−}}{a_{−}} − \frac{e_{−α}}{a_{+}} \\ 1 − d_{−}e_{−ν} + \frac{b_{−}d_{+}e_{−ν} −b_{−}e_{−λ}}{b_{+}} & 0 & \frac{D}{a_{−}b_{+}} \\ \frac{b_{−}e_{−(λ+α)}−b_{+}e_{−α}}{C} & 0 & − \frac{De_{−α}}{a_{−}C} \\ 0 & −a_{−}e_{−ν} & 1 + \frac{a_{−}d_{+}e_{−ν}}{a_{+}} \\ 0 & − \frac{a_{+}e_{−α}}{A} & \frac{d_{+}e_{−α}+e_{−2α}}{A} \end{matrix} ⌋ ,

Λ = diag[e_{α}, e_{−α}, e_{ν−α}, e_{α}, e_{−α}, e_{α−ν}]

if ν > α.

Observe that the condition a_{+}b_{−} = a_{−}b_{+} was not used in the above factorizations. Case 2: C = 0. From this and the equality a_{+}b_{−} = a_{−}b_{+} it follows that d_{−}b_{+} = d_{+}b_{−} = d_{−}a_{+} = d_{+}a_{−},

which allows us (by using elementary row and column operations) to suppose without loss of generality that d_{+} = d_{−} = 0. Under these additional conditions, the AP_{W} factorization of G is given by

G_{+} = [\begin{matrix}0 & 0 & e_{α} & −b_{−} & 0 & 0 \\ 0 & e_{λ} − \frac{a_{−}}{a_{+}} & 0 & 0 & − \frac{e_{λ}}{a_{−}} & −e_{ν} \\ − \frac{e_{ν}}{b_{−}} & \frac{a_{−}b_{−}f_{+}−a_{+}b_{+}f_{−}e_{λ}}{a_{+}b_{−}b_{+}} & − \frac{1}{b_{+}} & e_{ν} & \frac{f_{+}e_{λ}}{a_{−}b_{+}} & \frac{f_{+}e_{ν}}{b_{+}} \\ − \frac{b_{+}}{b_{−}} & \frac{De_{α}}{b_{−}} & 0 & b_{+} & 0 & 0 \\ 0 & 1 & 0 & 0 & − \frac{1}{a_{−}} & 0 \\ 0 & a_{+}e_{α} & 0 & 0 & − \frac{a_{+}e_{α}}{a_{−}} & −a_{+} \end{matrix} ] ,

G_{−} = [\begin{matrix}− \frac{b_{−}}{b_{+}} & − \frac{b_{−}f_{−}e_{−ν}}{b_{+}} & − \frac{b_{−}^{2}e_{−ν}}{b_{+}} & 1 − \frac{b_{−}e_{−λ}}{b_{+}} & 0 & \frac{D}{a_{−}b_{+}} \\ 0 & 1 & 0 & 0 & 0 & \frac{e_{−α}}{a_{−}} \\ 1 & f_{−}e_{−ν} & b_{−}e_{−ν} & e_{−λ} & 0 & 0 \\ 0 & \frac{f_{−}}{b_{−}} & 1 & \frac{e_{−α}}{b_{−}} & 0 & 0 \\ 0 & 0 & 0 & 0 & −a_{−}e_{−ν} & 1 \\ 0 & − \frac{a_{−}}{a_{+}} & 0 & 0 & 1 & − \frac{e_{−α}}{a_{+}} \end{matrix} ] ,

Λ = diag[1, 1, e_{ν}, e_{α}, e_{−α}, e_{−ν}].