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Matrix completions, norms and Hadamard products


Author: Roy Mathias
Journal: Proc. Amer. Math. Soc. 117 (1993), 905-918
MSC: Primary 15A60
DOI: https://doi.org/10.1090/S0002-9939-1993-1116267-7
MathSciNet review: 1116267
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Abstract: Let $ {M_{m,n}}$ (respectively, $ {H_n}$) denote the space of $ m \times n$ complex matrices (respectively, $ n \times n$ Hermitian matrices). Let $ S \subset {H_n}$ be a closed convex set. We obtain necessary and sufficient conditions for $ {X_0} \in S$ to attain the maximum in the following concave maximization problem:

$\displaystyle \max \{ {\lambda _{\min }}(A + X):X \in S\} $

where $ A \in {H_n}$ is a fixed matrix. Let $ \circ $ denote the Hadamard (entrywise) product, i.e., given matrices $ A = [{a_{ij}}],\;B = [{b_{ij}}] \in {M_{m,n}}$ we define $ A \circ B = [{a_{ij}}{b_{ij}}] \in {M_{m,n}}$.

Using the necessary and sufficient conditions mentioned above we give elementary and unified proofs of the following results. (a) For any $ A \in {M_{n,n}}$

$\displaystyle \omega (A) \equiv \max \{ \vert{x^{\ast}}Ax\vert:x \in {C^n},\;{x^{\ast}}x = 1\} \leqslant 1$

if and only if there is a matrix $ Z \in {H_n}$ such that

$\displaystyle \left( {\begin{array}{*{20}{c}} {I + Z} & A \\ {{A^{\ast}}} & {I - Z} \\ \end{array} } \right) \geqslant 0.$

(b) For any $ A \in {M_{m,n}}$

$\displaystyle \max \{ \vert\vert A \circ B\vert{\vert _\infty }:\vert\vert B\vert{\vert _\infty } \leqslant 1\} \leqslant 1$

if and only if there are matrices $ P \in {H_m},\;Q \in {H_n}$ such that

$\displaystyle \left( {\begin{array}{*{20}{c}} P & A \\ {{A^{\ast}}} & Q \\ \end... ... \right) \geqslant 0,\qquad P \circ I \leqslant I,\qquad Q \circ I \leqslant I.$

(c) For any $ A \in {M_{n,n}}$

$\displaystyle \max \{ \omega (A \circ B):\omega (B) \leqslant 1\} \leqslant 1$

if and only if there is a matrix $ P \in {H_n}$ such that

$\displaystyle \left( {\begin{array}{*{20}{c}} P & A \\ {{A^{\ast}}} & P \\ \end{array} } \right) \geqslant 0,\qquad P \circ I \leqslant I.$

We also consider other norms that can be represented in this way and show that if a norm can be represented in this way then so can its dual.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1116267-7
Keywords: Positive semidefinite matrix completion, Hadamard product, Schur product, numerical radius, convex optimization
Article copyright: © Copyright 1993 American Mathematical Society

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