Positive Semidefinite Hermitian Matrices Research Articles

Extreme rays and faces for the cone of class functions non-negative on the set of Gram matrices

If f is a C -valued function with domain S m , the symmetric group on {1,2,…, m}, then the matrix function [ f](·), or d f (·), is defined by [ f]( A)=∑ σ f( σ)∏ t=1 m a t, σ( t) for all m× m complex matrices A=[ a ij ]. We consider the cone K m cℓ whose elements are the Hermitian class functions f: S m→ C such that [ f]( A)⩾0 for each A∈ H m , where H m denotes the set of all m× m positive semi-definite Hermitian matrices. The extreme rays of K cℓ m are fundamental to an understanding of the linear inequalities that result by restricting the various [ f](·) to the sets H m . In particular, the resolution of the permanent dominance conjecture for immanants and certain related conjectures such as the conjectures of Lieb and Soules will likely involve identification and analysis of these rays. Barrett, Hall, and Loewy gave a complete list of the extreme rays of K cℓ m when m⩽4, and have shown that K 5 cℓ is not polyhedral. Given positive integers n and p such that n⩽ p and n+ p= m, we let K cℓ n, p denote the subcone of K m cℓ consisting of all f∈ K cℓ m such that f is expressible as a linear combination of the irreducible characters of S m associated with partitions of the form (2 i ,1 m−2 i ) where 0⩽ i⩽ n. We show that K cℓ n, p is an extreme polyhedral subcone, or face, of K cℓ m , and give explicit formulas for each of its n+1 extreme rays. Thus, K m cℓ has non-trivial polyhedral faces for all m.

Extreme rays for the large cone of hermitian generalized matrix functions

Given a <artwork name="GLMA31007ei1">-valued function f with domain <artwork name="GLMA31007ei2">, the symmetric group on {1,2,…, m}, we define the generalized matrix function [ f ](⋅), or df (⋅), in the usual way on the set of all m× m complex matrices. Letting <artwork name="GLMA31007ei3"> denote the set of all m× m positive semi-definite Hermitian matrices we consider the cone K m whose elements are the Hermitian functions <artwork name="GLMA31007ei4"> such that [ f ]( A)≥0 for all <artwork name="GLMA31007ei5">. The extreme rays in K m are fundamental to an understanding of the linear inequalities that result by restricting the generalized matrix functions [ f ](⋅) to the sets <artwork name="GLMA31007ei6">. In particular, the resolution of Lieb's permanent dominance conjecture, and certain similar conjectures such as the conjecture of Soules, will likely require identification and careful analysis of these rays. Grone, Merris, and Watkins have shown that the determinant function det(⋅), which is [ f ](⋅) if f is the signum function, is extreme in K m for each m. We identify additional rays that are extreme for all m. In particular, we associate with each 2-term partition <artwork name="GLMA31007ei7"> of {1,2,…, m} an element <artwork name="GLMA31007ei8"> that is shown to be extreme in K m for each m. If <artwork name="GLMA31007ei9"> is trivial, then <artwork name="GLMA31007ei10"> reduces to the determinant function; hence, our results are a natural extension of the result of Grone, Merris, and Watkins. Moreover <artwork name="GLMA31007ei11">, like det ( A), is expressible as a function of the eigenvalues of certain matrices related to A. Additional classes of extreme rays are also presented.

Two inequalities involving Hadamard products of positive semi-definite Hermitian matrices

We extend two inequalities involving Hadamard products of positive definite Hermitian matrices to positive semi-definite Hermitian matrices. Simultaneously, we also show the sufficient conditions for equalities to hold. Moreover, some other matrix inequalities are also obtained. Our results and methods are different from those which are obtained by S. Liu in [J. Math. Anal. Appl. 243:458-463(2000)] and B.-Y. Wang et al in [Lin. Alg. Appl. 302-303:163-172(1999)].

Multiplicative preservers on semigroups of matrices

We characterize multiplicative maps φ on semigroups of square matrices satisfying φ( P)⊆ P for matrix sets P , such as rank k (idempotent) matrices, totally nonnegative matrices, P 0 matrices, M 0 matrices, positive semidefinite matrices, Hermitian matrices, normal matrices, and contractions. We also characterize multiplicative maps φ satisfying φ( g( X))= φ( X) for various functions g on square matrices, such as the spectrum, spectral radius, numerical range, numerical radius, and matrix norms.

Complex Fractional Programming Involving Generalized Quasi/Pseudo Convex Functions

(P) minimize Re'f(z, z) + (z H Az) 1/2 ]/Re[g(z, z) - (z H Bz)1/2] subject to h(z, z) ∈ S ⊂ C m , z ∈ C m , where f, g: C 2n → C and h: C 2n → C m are analytic functions, A, B ∈ C m×n are positive semidefinite Hermitian matrices, S is a polyhedral cone in C m . In this paper, we establish conditions for the existence of an optimal solution in (P) involving (£, p, θ)-quasiconvex/-pseudoconvex analytic functions. Based on the sufficient optimality theorem, we construct a duality model and then establish weak/strong, and strict converse duality theorems.

Associative and Jordan Algebras, and Polynomial Time Interior-Point Algorithms for Symmetric Cones

We present a general framework whereby analysis of interior-point algorithms for semidefinite programming can be extended verbatim to optimization problems over all classes of symmetric cones derivable from associative algebras. In particular, such analyses are extendible to the cone of positive semidefinite Hermitian matrices with complex and quaternion entries, and to the Lorentz cone. We prove the case of the Lorentz cone by using the embedding of its associated Jordan algebra in the Clifford algebra. As an example of such extensions we take Monterio's polynomial-time complexity analysis of the family of similarly scaled directions—introduced by Monteiro and Zhang (1998)—and generalize it to cone-LP over all representable symmetric cones.

A nonpolyhedral cone of class function inequalities for positive semidefinite matrices

A function f from the symmetric group Sn into R is called a class function if it is constant on each conjugacy class. Let df be the generalized matrix function associated with f, mapping the n-by-n Hermitian matrices to R. For example, if f(σ)=sgn(σ), then df(A)=detA. Let Kn(Kn(R)) denote the closed convex cone of those f for which df(A)⩾0 for all n-by-n positive semidefinite Hermitian (real symmetric) matrices. For n=1,2,3,4 it is known that Kn and Kn(R) are polyhedral and there is a finite set of “test” matrices Tn(Tn(R)) such that f belongs to Kn(Kn(R)) if and only if df(A)⩾0 for each A in Tn(Tn(R)). We show here that K5 and K5(R) are not polyhedral. Thus, for n=5 there is no finite set of “test” matrices sufficient to establish which generalized matrix functions are nonnegative on the positive semidefinite matrices.

Hermitian positive semidefinite matrices whose entries are 0 or 1 in Modulus*

We show that a matrix is a Hermitian positive semidefinite matrix whose nonzero entries have modulus 1 if and only if it is similar to a direct sum of all I's matrices and a 0 matrix via a unitary monomial similarity. In particular, the only such nonsingular matrix is the identity matrix and the only such irreducible matrix is similar to an all l's matrix by means of a unitary diagonal similarity. Our results extend earlier results of Jain and Snyder for the case in which the nonzero entries (actually) equal 1. Our methods of proof, which reiy on the so called principal submatrix rank property, differ from the approach used by Jain and Snyder.

The Cone of Class Function Inequalities for the 4-By-4 Positive Semidefinite Matrices

A function f from the symmetric group Sn into R is called a class function if f(σ) = f(τ) whenever τ is conjugate to σ. Let df be thegeneralized matrix function associated with f, mapping the n-by-n positive semidefinite Hermitian matricesto R. For example, if f(σ) = sgn(σ), then df(A) = det A. We consider the cone Kn of those f for which df(A) ⩾ 0 for all n-by-n positive semidefinite Hermitian matrices. For n = 4 we show that Kn is polyhedral, and explicitly find the extreme rays. Equivalently, f belongs to Kn if df(A) ⩾ 0 for a finite minimal set of ‘test matrices’. This solves a problem posed by Gordon James. As a corollary, we characterize all inequalities involving linear combinations of immanants on the positive semidefinite Hermitian matrices of order n = 4. We obtain similar results for 4-by-4 real positive semidefinite matrices. 1991 Mathematics Subject Classification: 15A15, 15A45, 15A48.

Tensor inequalities, ξ-functions and inequalities involving immanants

If f is a complex valued function with domain S n , the symmetric group on {1,2,…, n}, then we define the matrix function d f (·) in the usual way. If α is a partition of n and λ α denotes the associated irreducible character of S n , then d λα (·), which we abbreviate to d α (·), is known as an immanant. The permanent dominance conjecture of Lieb specialized to the symmetric groups asserts that if α is a partition of n, then d α(A)⩽ deg(α) per (A) for each A∈ H n , the set of n× n positive semi-definite Hermitian matrices. Recently it was shown (T.H. Pate, Proc. London Math. Soc. 76 (2) (1998) 307–358.) that if α is a partition of n of the form ( p, q, r,2 s ,1 t ), then d α ( A)⩽deg( α)per( A) for each A∈ H n . We improve upon this result by showing that this same inequality holds for all A∈ H n when α is a partition of the form ( p, q w , r,2 s ,1 t ) where 0⩽ w⩽2. Thus, permanent dominance holds for all immanants whose associated partitions are of the form ( p, q 2, r). We also show that permanent dominance holds for all immanants whose associated partitions are of the form ( n+ p, n k ) as long as n is sufficiently large. Inequalities involving immanants have often been obtained using a special kind of class function known as a Ψ-function. Our results on immanants are obtained by analyzing a new set of class functions, called ξ-functions, that are more fundamental than the Ψ-functions in the sense that each Ψ-function is a non-negative linear combination of ξ-functions. The ξ-functions arise from a study of tensor contractions and a special collection of quadratic forms defined on spaces of bi-symmetric tensors.

Some inequalities for sum and product of positive semidefinite matrices

The purpose of this paper is to present some inequalities on majorization, unitarily invariant norm, trace, and eigenvalue for sum and product of positive semidefinite (Hermitian) matrices. Some open questions proposed by Marshall and Olkin are resolved.

Extreme rays for cones of Hermitian matrix functions that are non-negative on the positive semi-definite matrices

Given a complex valued function λ with domain s n , the symmetric group on {1,2,…, n ), we define the matrix function d λ (·) in the usual way, and restrict d λ (·) to ℋ n , the n × n positive semi-definite Hermitian matrices. If λ∈ℂS n ,. the algebra of functions from S n to ℂ. then λ is said to be Hermitian if λ(σ −1 = λ(σ) for each σ ∈ S n . By K n we mean the cone of Hermitian elements λ ∈ℂS n such that d λ {A) ⩾0 for each A ∈ℋ n , and by K el n , also a cone, we mean the intersection of K n with the set of class functions on S n . Recently Barrett, Hall, and Loewy showed that if n ⩽4. then K el n is finitely generated, and that there is a finite set ℐ.U n ⊂ ∛ n of matrices, called test matrices, such that a class function λ: S n → ℂ is in K el n if and only if d λ { A ) ⩾ 0 for each A ∈ ℐ.U n . We demonstrate that if n ⩾ 3. then the larger cone K n is not finitely generated by presenting an infinite set of extreme rays. Consequently, there is no finite set of test matrices for K n when n ⩾ 3. In addition we present a scheme that is effective in identifying extreme rays in both K n and K ef n and use it in conjunction with various sets of test matrices to identify additional extreme rays in each of these cones. In particular, certain of the rays associated with the Fischer inequality are proved to be extreme in K n , and K ef n .

Some inequalities for singular values and eigenvalues of generalized Schur complements of products of matrices

In this paper, using transformation of Schur complements of matrices and some estimates of eigenvalues of positive semidefinite Hermitian matrices, we obtain some inequalities of singular values for Schur complements of products of matrices.

Hook immanantal inequalities for trees explained

Let d ¯ k denote the normalized hook immanant corresponding to the partition ( k , l n-k ) of n . P. Heyfron proved the family of immanantal inequalities det A = d ¯ 1 ( A ) ⩽ d ¯ 2 ( A ) ⩽ ⋯ ⩽ d ¯ n ( A ) = per A ( 1 ) for all positive semidefinite Hermitian matrices A. Motivated by a conjecture of R. Merris, it was shown by the authors that (1) may be improved to d ¯ k − 1 ( L ( T ) ) ⩽ k − 2 k − 1 d ¯ k ( L ( T ) ) ( 2 ) for all 2 ⩽ k ⩽ n whenever L ( T ) is the Laplacian matrix of a tree T . The proof of (2) relied on rather involved recursive relations for weighted matchings in the tree T as well as identities of hook characters. In this work, we circumvent this tedium with a new proof using the notion of vertex orientations. This approach makes (2) immediately apparent and more importantly provides an insight into why it holds, namely the absence of certain vertex orientations for all trees. As a by-product we obtain an improved bound, 0 ⩽ 1 k − 1 [ d ¯ k ( L ( T )) − d ¯ k ( L ( S ( n ) ) ) ] ⩽ k − 2 k − 1 d ¯ k ( L ( T ) ) − d ¯ k − 1 ( L ( T ) ) , where S ( n ) is the star with n vertices. The ease with which the inequality in (2) and its improvement are derived points to the value of the concept of vertex orientation in the study of immanantal inequalities on graphs.

Row Appending Maps, Ψ-Functions, and Immanant Inequalities for Hermitian Positive Semi-Definite Matrices

Row Appending Maps, Ψ-Functions, and Immanant Inequalities for Hermitian Positive Semi-Definite Matrices

Hook immanantal inequalities for Laplacians of trees

For an irreducible character X λ of the symmetric group S n , indexed by the partition λ, the immanant function d λ , acting on an n × n matrix A = ( a ij ), is defined as d λ ( A) = Σ σ ∈ S n X λ ( σ) Π n i = 1 a iσ( i) . The associated normalized immanant d ̄ λ is defined as d ̄ λ = d λ X λ(identity) where identity is the identity permutation. P. Heyfron has shown that for the partitions ( k, 1 n− k ), the normalized immanant d ̄ k satisfies det A = d 1( A) ⩽ d 2( A) ⩽ ⋯ ⩽ d n ( A) = per A (1) for all positive semidefinite Hermitian matrices A. When A is restricted to the Laplacian matrices of graphs, improvements on the inequalities above may be expected. Indeed, in a recent survey paper, R. Merris conjectured that d n−1 ( A) ⩽ n − 2 n − 1 d n ( A) (2) wherever A is the Laplacian matrix of a tree. In this note, we establish a refinement for the family of inequalities in (1) when A is the Laplacian matrix of a tree, that includes (2) as a special case. These inequalities are sharp and equality holds if and only if A is the Laplacian matrix of the star. This is proved via the inequalities d ̄ k(A) − d ̄ k − 1(A) ⩽ d ̄ k + 1(A) − d ̄ k(A) for k = 2, 3, …, n − 1, where A is the Laplacian matrix of a tree.

Hook immanantal inequalities for Laplacians of trees

For an irreducible character X λ of the symmetric group S n , indexed by the partition λ, the immanant function d λ , acting on an n × n matrix A = ( a ij ), is defined as d λ ( A ) = Σ σ ∈ S n X λ ( σ ) Π n i = 1 a iσ ( i ) . The associated normalized immanant d ̄ λ is defined as d ̄ λ = d λ X λ (identity) where identity is the identity permutation. P. Heyfron has shown that for the partitions ( k , 1 n − k ), the normalized immanant d ̄ k satisfies det A = d 1 ( A ) ⩽ d 2 ( A ) ⩽ ⋯ ⩽ d n ( A ) = per A (1) for all positive semidefinite Hermitian matrices A . When A is restricted to the Laplacian matrices of graphs, improvements on the inequalities above may be expected. Indeed, in a recent survey paper, R. Merris conjectured that d n −1 ( A ) ⩽ n − 2 n − 1 d n ( A ) (2) wherever A is the Laplacian matrix of a tree. In this note, we establish a refinement for the family of inequalities in (1) when A is the Laplacian matrix of a tree, that includes (2) as a special case. These inequalities are sharp and equality holds if and only if A is the Laplacian matrix of the star. This is proved via the inequalities d ̄ k (A) − d ̄ k − 1 (A) ⩽ d ̄ k + 1 (A) − d ̄ k (A) for k = 2, 3, …, n − 1, where A is the Laplacian matrix of a tree.

A partial ordering of CS m defined by the generalized matrix functions

We present a description of a partial ordering of the complex full symmetric group algebra, CS m , via generalized matrix functions, d f ( A), defined on the set of all m × m complex matrices A. We show that for f:S m → C, if d f ( A) = 0 for all positive semidefinite Hermitian matrices A, then f = 0. Thus we can build up a partial ordering for CS m .

Exterior products, elementary symmetric functions, and the Fischer determinant inequality

If n and k are positive integers such that k < n, and A = [ a ij ] is an n × n complex matrix, then let A [ k] denote the k × k principal submatrix [ a ij ] k i, j=1 , and let A ( k) denote the corresponding complementary principal submatrix. The inequality of Fischer guarantees us that if A ∈ H n, the n × n positive semidefinite Hermitian matrices, then det A ⩽ det A [ k] det A ( k) for k, 1 ⩽ k < n. We present a careful analysis of the Fischer error function E k(A) = det A [k] det A (k) — det A that is based on certain geometric invariants. This analysis leads to various expansion formulas and a matrix theoretic refinement and extension of the Fischer inequality. In particular, for each k such that 1 ⩽ k < n we present k + 1 matrix functions ψ 0, k , ψ 1, k , …, ψ k, k , such that if A ∈ H n, then ψ 0, k ( A) = det A, ψ k, k ( A) = det A [ k] det A ( k) , and ψ i, k ( A) ⩽ ψ i+1, k ( A) for each i such that 0 ⩽ i ⩽ k − 1. As we demonstrate, the matrix functions ψ 0, k , ψ 1, k , …, ψ k, k are closely related to the various Laplace expansion formulas for the determinant function. For example, the inequality ψ 0, k ( A) ⩽ ψ 1, k ( A), A ∈ H n, translates into det A ⩽ ( 1 k )∑ k i, j=1(−1) i+ja ij det A(i|j) , where A = [ a ij ] and A( i| j) denotes A with row i and column j deleted. As a bonus we obtain inequalities for the higher order differences of the sequences ψ 0, k ( A), ψ 1, k ( A), …, ψ k, k ( A) where A ∈ H n. Specifically, we define the difference operator ∇ according to ∇ x i = x i − x i+1 , and show that if A ∈ H n and q + t ⩽ k, then (−1) q ∇ q ψ t, k ( A) ⩾ 0.

A machine for producing inequalities involving immanants and other generalized matrix functions

There are many known inequalities involving the restriction of immanants and other generalized matrix functions to the set of positive semidefinite Hermitian matrices. We present a general scheme for obtaining such inequalities. Matrix theoretic refinements of Lieb's and Fischer's inequalities, recently discovered by Pate, are obtainable using the scheme. Other known inequalities are discussed, and we derive new inequalities that relate characters constructed from generalized diagrams.