A class of Hessenberg matrices with known pseudoinverse and Drazin inverse
Authors:
Inderjit Singh, George Poole and Thomas Boullion
Journal:
Math. Comp. 29 (1975), 615619
MSC:
Primary 65F20
MathSciNet review:
0368407
Fulltext PDF Free Access
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Abstract: In this paper, a class of Hessenberg matrices is presented for adoption as test matrices. The MoorePenrose inverse and the Drazin inverse for each member of this class are determined explicitly.
 [1]
Adi
BenIsrael, An iterative method for computing the
generalized inverse of an arbitrary matrix, Math. Comp. 19 (1965), 452–455. MR 0179915
(31 #4152), http://dx.doi.org/10.1090/S00255718196501799155
 [2]
T.
S. Chow, A class of Hessenberg matrices with known eigenvalues and
inverses., SIAM Rev. 11 (1969), 391–395. MR 0252407
(40 #5627)
 [3]
Henry
P. Decell Jr., An application of the CayleyHamilton theorem to
generalized matrix inversion, SIAM Rev. 7 (1965),
526–528. MR 0194446
(33 #2656)
 [4]
Graeme
Fairweather, On the eigenvalues and eigenvectors of a class of
Hessenberg matrices., SIAM Rev. 13 (1971),
220–221. MR 0286807
(44 #4016)
 [5]
Magnus
R. Hestenes, Inversion of matrices by biorthogonalization and
related results, J. Soc. Indust. Appl. Math. 6
(1958), 51–90. MR 0092215
(19,1080d)
 [6]
Jo
Ann Howell and Robert
T. Gregory, An algorithm for solving linear algebraic equations
using residue arithmetic. I, Nordisk Tidskr. Informationsbehandling
(BIT) 9 (1969), 200–224. MR 0261775
(41 #6388a)
 [7]
Carl
D. Meyer Jr., Limits and the index of a square matrix, SIAM J.
Appl. Math. 26 (1974), 469–478. MR 0364284
(51 #539)
 [8]
Morris
Newman, Integral matrices, Academic Press, New York, 1972.
Pure and Applied Mathematics, Vol. 45. MR 0340283
(49 #5038)
 [9]
G.
Poole and T.
Boullion, The Drazin inverse and a spectral inequality of Marcus,
Minc, and Moyls, J. Optimization Theory Appl. 15
(1975), 503–508. MR 0366950
(51 #3196)
 [10]
L.
Duane Pyle, Generalized inverse computations using the gradient
projection method, J. Assoc. Comput. Mach. 11 (1964),
422–428. MR 0172451
(30 #2670)
 [11]
C.
Radhakrishna Rao and Sujit
Kumar Mitra, Generalized inverse of matrices and its
applications, John Wiley\thinspace&\thinspace Sons, Inc., New
YorkLondonSydney, 1971. MR 0338013
(49 #2780)
 [12]
W.
T. Stallings and T.
L. Boullion, Computation of pseudoinverse matrices using residue
arithmetic, SIAM Rev. 14 (1972), 152–163. MR 0307456
(46 #6576)
 [1]
 A. BENISRAEL, "An iterative method for computing the generalized inverse of an arbitrary matrix," Math. Comp., v. 19, 1965, pp. 452455. MR 31 #4152. MR 0179915 (31:4152)
 [2]
 T. S. CHOW, "A class of Hessenberg matrices with known eigenvalues and inverses," SIAM Rev., v. 11, 1969, pp. 391395. MR 40 #5627. MR 0252407 (40:5627)
 [3]
 H. P. DECELL, "An application of the CayleyHamilton theorem to generalized matrix inversion," SIAM Rev., v. 7, 1965, pp. 526528. MR 33 #2656. MR 0194446 (33:2656)
 [4]
 G. FAIRWEATHER, "On the eigenvalues and eigenvectors of a class of Hessenberg matrices," SIAM Rev., v. 13, 1971, pp. 220221. MR 44 #4016. MR 0286807 (44:4016)
 [5]
 M. R. HESTENES, "Inversion of matrices by biorthogonalization and related results," J. Soc. Indust. Appl. Math., v. 6, 1958, pp. 5190. MR 19, 1080. MR 0092215 (19:1080d)
 [6]
 J. A. HOWELL & R. T. GREGORY, "An algorithm for solving linear algebraic equations using residue arithmetic. I," BIT, v. 9, 1969, pp. 200224. MR 41 #6388a. MR 0261775 (41:6388a)
 [7]
 C. D. MEYER, "Limits and the index of a square matrix," SIAM J. Appl. Math., v. 26, 1974, pp. 469478. MR 0364284 (51:539)
 [8]
 M. NEWMAN, Integral Matrices, Academic Press, New York, 1972. MR 0340283 (49:5038)
 [9]
 G. POOLE & T. BOULLION, "The Drazin inverse and a spectral inequality of Marcus, Minc and Moyls," J. Optimization Theory Appl. (To appear.) MR 0366950 (51:3196)
 [10]
 L. D. PYLE, "Generalized inverse computations using the gradient projection method," J. Assoc. Comput. Mach., v. 11, 1964, pp. 422428. MR 30 #2670. MR 0172451 (30:2670)
 [11]
 C. RAO & S. MITRA, Generalized Inverse of Matrices and Applications, Wiley, New York, 1971. MR 0338013 (49:2780)
 [12]
 W. STALLINGS & T. BOULLION, "Computation of pseudoinverse matrices using residue arithmetic," SIAM Rev., v. 14, 1972, pp. 152163. MR 0307456 (46:6576)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197503684072
PII:
S 00255718(1975)03684072
Keywords:
Hessenberg matrix,
pseudoinverse,
Drazin inverse
Article copyright:
© Copyright 1975 American Mathematical Society
