Positive Semidefinite Hermitian Research Articles

On a Conjecture of Pierce for Permanents of Singular Correlation Matrices

Let $I_n $ be the n-by-n identity matrix, let $J_n $, be the n-by-n matrix, all of whose entries equal 1, and let $Y_{n = } n/( n - 1 ) I_n - 1/( n - 1 ) J_n $. $Y_n $ is then a singular correlatio...

On a conjecture of h. minc∗

It is shown that if A,B are n × n positive semidefinite Hermitian matrices, then necessary and sufficient condition is given for equality to hold.

Tensor products, symmetric products, and permanents of positive semi-definite Hermitian matrices

Let S k(C n) denote the symmetric complex valued k-linear functions on (C n)k. We prove the inequality where A,C ∈ S n (C m) and B,D ∈ S p(C m). Equality results if and only if there exist λ,u ∈ C such that A=λC and B= μD. If w 1,w 2,…,wk are in C m then G(w 1,w 2,…,wk ) denotes the Gram matrix associated with w 1,w 2,…,wk An application of the above inequality to decomposables yields the inequality where and

Some inequalities for the eigenvalues of the product of positive semidefinite Hermitian matrices

Let λ 1( A)⩾⋯⩾ λ n ( A) denote the eigenvalues of a Hermitian n by n matrix A, and let 1⩽ i 1< ⋯ < i k ⩽ n. Our main results are ∑ t=1 k λ t (GH)⩽ ∑ t=1 k λ i t(G)λ n−i t +1(H) and ∑ t=1 k λ i t (GH)⩽ ∑ t=1 k λ i t(G)λ n−t+1(H) . Here G and H are n by n positive semidefinite Hermitian matrices. These results extend Marshall and Olkin's inequality ∑ t=1 k λ t (GH)⩽ ∑ t=1 k λ t(G)λ n−t+1(H) . We also present analogous results for singular values.

Partitions, irreducible characters, and inequalities for generalized matrix functions

Given a partition $\alpha = \{ {\alpha _1},{\alpha _2}, \ldots ,{\alpha _s}\}$, ${\alpha _1} \geq {\alpha _2} \geq \cdots \geq {\alpha _s}$, of $n$ we let ${X_\alpha }$ denote the derived irreducible character of ${S_n}$, and we associate with $\alpha$ a derived partition \[ \alpha \prime = \{ {\alpha _1} - 1,{\alpha _2} - 1, \ldots ,{\alpha _t} - 1,{\alpha _{t + 1}}, \ldots ,{\alpha _s},{1^t}\} \] where $t$ denotes the smallest positive integer such that ${\alpha _t} > {\alpha _{t + 1}}\;({\alpha _{s + 1}} = 0)$. We show that if $Y$ is a decomposable $\mathbb {C}$-valued $n$-linear function on ${\mathbb {C}^m} \times {\mathbb {C}^m} \times \cdots \times {\mathbb {C}^m}$ ($n$-copies) then $\left \langle {{X_\alpha }Y,Y} \right \rangle \geq \left \langle {{X_\alpha },Y,Y} \right \rangle$. Translating into the notation of matrix theory we obtain an inequality involving the generalized matrix functions ${d_{{X_\alpha }}}$ and ${d_{{X_{\alpha \prime }}}}$, namely that \[ {({X_\alpha }(e))^{ - 1}}{d_{{X_\alpha }}}(B) \geq {({X_{\alpha \prime }}(e))^{ - 1}}{d_{{X_{\alpha \prime }}}}(B)\] for each $n \times n$ positive semidefinite Hermitian matrix $B$. This result generalizes a classical result of I. Schur and includes many other known inequalities as special cases.

Hadamard Square Roots

If A is an n-square positive semidefinite Hermitian matrix of rank 1, then the Hadamard square root of A is the n-square matrix obtained by replacing each entry of A by the principal value of its square root. It is proved that if A has no zero or negative entries, then the Hadamard square root has odd rank and all odd ranks are possible.

Fast Polar Decomposition of an Arbitrary Matrix

The polar decomposition of an $m \times n$ matrix A of full rank, where $m \geqq n$, can be computed using a quadratically convergent algorithm of Higham [SIAM J. Sci. Statist. Comput., 7(1986), pp. 1160–1174]. The algorithm is based on a Newton iteration involving a matrix inverse. It is shown how, with the use of a preliminary complete orthogonal decomposition, the algorithm can be extended to arbitrary A. The use of the algorithm to compute the positive semidefinite square root of a Hermitian positive semidefinite matrix is also described. A hybrid algorithm that adaptively switches from the matrix inversion based iteration to a matrix multiplication based iteration due to Kovarik, and to Björck and Bowie, is formulated. The decision when to switch is made using a condition estimator. This “matrix multiplication rich” algorithm is shown to be more efficient on machines for which matrix multiplication can be executed 1.5 times faster than matrix inversion.

An alternate variational characterization of matrix Riccati equation solutions

A new variational characterization of the solution of a matrix Riccati equation is presented. The characterization is given as the maximum Hermitian matrix that makes a certain compound matrix Hermitian positive semidefinite. The shorted operator (Schur complement) applied to the compound matrix produces the Riccati equation.

Extremal bipartite matrices

Let G be an undirected graph on vertices {1,…, n }. Let M ( G ) be the convex cone of all positive semidefinite hermitian matrices A satisfying a ij = 0 if ( i, j ) is not an edge of G . In the case that G is a complete bipartite graph, we characterize all extreme rays of M ( G ). In doing so, we shall also find the maximum rank that an extreme matrix in M ( G ) can have.

Semidefinite perturbations of analytic Hermitian matrix functions

In this paper we consider the behaviour of a real eigenvalue of an analytic Hermitian matrix valued function under perturbation with a positive semidefinite analytic Hermitian matrix valued function. We extend previous results on perturbation with positive definite functions to the positive semidefinite case.

The Ranks of Extremal Positive Semidefinite Matrices with Given Sparsity Pattern

Let P be a symmetric set of ordered pairs of integers from 1 to n, and define $M^ + (P)$ to be the closed cone of all positive semidefinite Hermitian matrices whose $(i,j)$ entry is zero whenever $i \ne j$ and $(i,j)$ is not in P. The extreme points of $M^ + (P)$ are considered. In some special cases, the maximum rank that such an extreme point can have is calculated.

OnH-contractions and the extension problem for hermitian block toeplitz matrices

It is well known that any positive semi-definite Hermitian block Toeplitz matrix Tn (C0 ,C1 ,…Cn ) has positive semi-definite Toeplitz extensions Tn+1 (C0 ,…Cn Cn+1 ). In this paper, we consider the more general problem of extending arbitrary Hermitian block Toeplitz matrices Tn to matrices Tn+1 with the same number of negative eigenvalues as Tn . A necessary and sufficient condition for the existence of such Cn+1 is derived, and a parametrization of all corresponding Cn+1 is presented. Our approach to the Hermitian block Toeplitz extension problem is based on a formulation as a completion problem for contractions in indefinite inner product spaces. A complete solution of this more general problem is given.

Lower bounds for permanents of Gram matrices having a rank one principal submatrix

Let A=[ a ij ] be a positive semidefinite Hermitian ( n+ p×( n+ p) matrix partitioned into blocks A> 11, A 12, A 21 and A 22 such that A 11 is n× n and A 22 is p× p. Let à denote the ( n+ p−1)×( n+ p−1) matrix obtained from A by deleting the first row and column of A. We prove that if A 11 is of rank one, then per(A)⩾na 11 per( A ̃ ) with equality if and only if A 12 is the zero matrix or A has a row of zeros.

Generalizing the inequality per(J2⊗M)⩾2n[per(M

Let M denote an n × n positive semidefinite Hermitian matrix,and let W = [ω ij ] be either a 2×2 real matrix such that ω 11 ω 22 ⩽0 and ω 12 ω 21 ⩽0 or a 2×2 Hermitian matrix such that ω 11 ω 22 ⩽0. We prove the inequality per (W⊗M)⩾[ per (W)] n [ per (M)] 2 and establish that equality results if and only if ω 11 ω 22 = 0 or ω 12 ω 21 =0 or M contains a row of zeros or M is diagonal.

Permanental Inequalities for Correlation Matrices

Let A be a positive semidefinite Hermitian matrix of order n with $|a_{11} | = \cdots = |a_{nn} | = 1$. We prove that $\operatorname{per} ( A )\geqq (1/n ) \| A \|^2 $, where $\| A \|$ is the Frobenius norm of A. This follows from a stronger result when $n = 4$, namely per $( A )\geqq \frac{1}{3} ( \| A \|^2 - 1 )$. Various corollaries are obtained.

Inequalities for single-hook immanants

Let d̄ k denote the normalized generalized matrix function based upon the irreducible character of S n associated with the partition ( k ,1,…,1) of n . For all positive semidefinite Hermitian n -by- n A , we prove certain inequalities in the chain det A= d ̄ 1 (A)⩽ d ̄ 2 (A)⩽⋯⩽ d ̄ n−1 (A)⩽ d ̄ n (A)= per A , conjectured for each n . This work is in pursuit of the “permanent on top“ conjecture, and this conjecture is also verified for another class of immanants. The methods involve several new inequalities for eigenvalues and the permanent which may be of independent interest.

Oppenheim's Inequality for the Second Immanant

AbstractDenote by d2 the immanant afforded by Sn and the character corresponding to the partition (2, 1n-2). If n ≥ 4, the following analog of Oppenheim's inequality is proved:for all n-by-n positive semidefinite hermitian A and B.

Generalizing the Fischer inequality

If q ⩾ 1, let A q denote the complex vector space whose elements are the complex-valued alternating q-linear functions on C m . If A ∈ A n , B ∈ A p are decomposable, then the classical Fischer inequality for partitioned positive semidefinite Hermitian matrices implies that n + p n ‖ A ⋀B‖ 2⩽‖A‖ 2·‖B‖ 2 . We show that this inequality holds if only one of A and B is decomposable or if A ∧ B is decomposable. We introduce the concept of relative decomposability and show that the above inequality continues to hold if A is decomposable relative to B. Also, an example is presented to demonstrate that the above inequality does not hold in general.

A Complex Orthogonal-Symmetric Analog of the Polar Decomposition

Every square complex matrix A can be factorized as $A = UH$, where U is unitary and H is positive semi-definite Hermitian. If A is nonsingular, it is known that one may write $A = QS$, where Q is complex orthogonal and S is complex symmetric. We develop necessary and sufficient conditions for there to be a factorization of this type when A is singular, and we give sufficient conditions for there to be at least one such factorization in which the factors commute.

A Fischer inequality for the second immanant

Denote by H n the cone of n-by- n positive semidefinite Hermitian matrices. Let d 2 be the generalized matrix function (or immanant) afforded by the symmetric group S n and the irreducible degree n − 1 character corresponding to the partition (2,1,…,1). Suppose A ∈ H n is partitioned into blocks, A= A 11 A 12 A ∗ 12 A 22 If n⩾4, then d 2(diag(A 11,A 22))⩾ d 2(A). This follows from the fact that d 2 is a Schur-concave function of the spectrum of A for A ∈ C n , where C n consists of the matrices in H n whose diagonal entries all equal 1.