Search

Given n×n Hermitian matrices, H1,…,Hp, a complete description is found for the possible inertias of the sum H1+⋯+Hp, in terms of the inertias of Hi, i=1,…,p. We also consider and solve other related problems.

Using elementary matrix algebra we establish the following theorems: (1.3) Let H be any n×n Hermitian matrix and let M be any n×m complex matrix. Suppose that (i) M∗M has r eigenvalues in the interval [a1, b1]; (ii) H has s eigenvalues in [a2, b2, a2⩾0. Then M∗HM has at least r+s−n eigenvalues in [a1a2, b1b2]. (3.1) Let H be any n×n Hermitian matrix with In H=(π, ν, δ). Let M be any real n×m matrix, and let δM=DimKer M. Let (π1, ν1, δ1) denote the inertia of M∗HM. Then π+(m −n)−δM⩽π1⩽π and ν+(m−n)−δM⩽ν1⩽ ν . When M is a square matrix, these inequalities are simply π−δM⩽π1⩽π and ν−δM⩽ν1⩽ν .

Some trace inequalities for Hermitian matrices and matrix products involving Hermitian matrices are presented.

If ( A , B ) is any pair of Hermitian matrices, the power of λ dividing det( λI − xA − yB ) will be given by the number of basic singular summands in the pair. Contrary to conjecture, this power can be greater than one even when the pair is unitarily irreducible.

Centrohermitian and skew-centrohermitian matrices are defined in analogy to centrosymmetric and skew-centrosymmetric matrices. The main results of this paper is that each square centrohermitian (skew-centrohermitian) matrix is similar to a matrix with real (pure imaginary) entries.

Unitary groups of Hermitian forms with a hyperbolic rank at least one over local rings have been studied by D. G. James (J. Algebra 52 (1978), 354–363). Using his methods, we extend some of his results, i.e., generators, congruence subgroups and the classification of subgroups normalized by the Eichler subgroups to more general rings which contain semilocal rings.

Let A and B be n × n Hermitian matrices. The matrix pair ( A , B ) is called definite pair and the corresponding eigenvalue problem β Ax = α Bx is definite if c ( A , B ) ≡ inf ‖ x ‖= 1 {| H ( A + iB ) x |} > 0. In this note we develop a uniform upper bound for differences of corresponding eigenvalues of two definite pairs and so improve a result which is obtained by G.W. Stewart [2]. Moreover, we prove that this upper bound is a projective metric in the set of n × n definite pairs.

The cone CP n,q of completely positive linear transformations from M n ( C )= M n to M q is shown to be isometrically isomorphic to P nq , the cone of nq by nq positive semidefinite matrices. Generalizations of scalar and matrix results to CP n, q ⊂ HP n, q ⊂ L ( M n , M q ) (where HP n,q represents the hermitian-preserving linear transformations) are discussed. Relationships among the completely positives, the set of positive semidefinite preservers π( P n ) , and its dual π( M n ) ∗ are given. Left and right facial ideals of CP are characterized. Properties of the joint angular field of values of a finite sequence of hermitian matrices H 1 ,…, H m are studied, leading to a characterization of π( P q , P n ) .

Let A be an n × n matrix; write A = H + iK , where i 2 =—1 and H and K are Hermitian. Let f ( x , y , z ) = det( zI − xH − yK ). We first show that a pair of matrices over an algebraically closed field, which satisfy quadratic polynomials, can be put into block, upper triangular form, with diagonal blocks of size 1×1 or 2×2, via a simultaneous similarity. This is used to prove that if f(x,y,z) = [g(x,y,z)] n 2 , where g has degree 2, then for some unitary matrix U , the matrix U ∗ AU is the direct sum of n 2 copies of a 2×2 matrix A 1 , where A 1 is determined, up to unitary similarity, by the polynomial g ( x , y , z ). We use the connection between f ( x , y , z ) and the numerical range of A to investigate the case where f ( x , y , z ) has the form ( z − αax − βy ) r [ g ( x , y , z )] s , where g ( x , y , z ) is irreducible of degree 2.

If v is a norm on C n, let H(v) denote the set of all norm-Hermitians in C nn. Let S be a subset of the set of real diagonal matrices D. Then there exists a norm v such that S=H(v) (or S = H(v)∩D) if and only if S contains the identity and S is a subspace of D with a basis consisting of rational vectors. As a corollary, it is shown that, for a diagonable matrix h with distinct eigenvalues λ1,…, λr, r⩽n, there is a norm v such that h ∈ H(v), but hs∉H(v), for some integer s, if and only if λ2–λ1,…, λr–λ1 are linearly dependent over the rationals. It is also shown that the set of all norms v, for which H(v) consists of all real multiples of the identity, is an open, dense subset, in a natural metric, of the set of all norms.

By using Hadamard products we give some reasonable upper and lower bounds of Golden-Thompson type for ∥ e H 1 +…+ H m ∥, where H i ( i = 1, 2, …, m ) are arbitrary Hermitian matrices and ∥·∥ is an arbitrary unitarily invariant norm.

Let A(z) be an n × n complex matrix whose elements depend analytically on z ∈ C m. It is well known that any individual eigenvalue of A(z) may be nondifferentiable when it coalesces with others. In this paper, we investigate the analycity property of functions on the eigenvalues λ(z) = (λ1(z),…, λn(z)) of A(z). We first introduce the notion of functions that are symmetric with respect to partitions. It is then shown that if a function ƒ : C n → C is analytic at λ(a), where a ϵ C m, and is symmetric with respect to a certain partition induced by λ(a), then the composite function g(z) = ƒ(λ 1 (z),…,λ n (z)) is analytic at a. When z is real, A(z) is symmetric or Hermitian, and the aforementioned assumptions hold, so that g(z) is analytic at a, we also derive formulae for its first and second order partial derivatives. We apply the results to several problems involving eigenvalues.

In 1980, Khatri (Linear Alg. Appl. 33 (1980) 57–65) has shown that the Hermitian part (A +A*)/2 of a square complex matrixA is idempotent and has the same rank asA if and only ifA is normal and the real part of any of its non-trivial eigenvalues is equal to one. In this note we investigate idempotency of (A +A*)/2 without assumptions on its rank and demonstrate thatA is necessarily an EP matrix, whereA is called EP if the range ofA equals the range ofA*.

Optimization involving eigenvalues arise in many engineering problems. We propose a new algorithm on minimizing the largest eigenvalue over an affine family of symmetric matrices. Under certain assumptions it is shown that, if started close enough to the minimizer x ∗ , the proposed algorithm converges to x ∗ quadratically. The proposed algorithm can be readily extended to minimizing the largest eigenvalue over an affine family of Hermitian matrices. Also, it has been extended to minimizing sums of the largest eigenvalues of a symmetric or Hermitian matrix.

A Hermitian design H(q) consists of the points and Hermitian unitals of PG(2,q2). A committee of H(q) is a blocking set of H(q) of minimum cardinality b(H(q)). It is proved that the committees of H(3) are the lines of PG(2, 9) and, for all odd q, that 2q + 2 ≤b(H(q)) < (1 + 7 ln q)(q2 + 1)q−1.

The concept of a strictly positive definite set of Hermitian matrices is introduced. It is shown that a strictly positive definite set is always a positive definite set, and conditions are found under which a positive definite set is strictly positive definite. We also show that a set of Hermitian matrices is strictly positive definite if and only if some nonnegative linear combination of these matrices is a positive definite matrix. For state dimension two, we use this concept to find necessary and sufficient conditions for a two-mode completely controllable irreducible multimodal system to be contractible relative to an elliptic norm. For general state dimensions, we give necessary and sufficient conditions for a special-type two-mode completely controllable irreducible system to be contractible relative to a weakly monotone norm. Applying the above results, we show that, for state dimension two, there exists a completely controllable two-mode system which is not contractible relative to either an elliptic or a weakly monotone norm. We leave open the question whether or not complete controllability implies contractibility, relative to some norm, for multimodal systems of two or more modes.

We describe how character values of certain Hecke algebras can be used to give linear combinations of immanants which are guaranteed to be positive on all Hermitian positive semidefinite matrices A . As an application, we present a short proof of a theorem of James and Liebeck that the permanent of A is greater than the normalized immanants which correspond to two-part partitions.

We prove a result on the existence of a common totally isotropic subspace for a pair of forms on V × V , where V is a finite dimensional space (real, complex, or quaternionic), and where one of the forms is hermitian while the other is skew-hermitian. Applications to algebraic matrix Riccati equations are given.

We discuss two comparison theorems for algebraic Riccati equations of the form XBR -1 B ∗ X−X(A−BR -1 C) -(A−BR -1 C) ∗ X−(Q−C ∗ R -1 C) = 0. Simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a Riccati equation from the existence of a hermitian solution of a second Riccati equation. Further, similar results are given for discrete algebraic Riccati equations.

Let M be the complex linear space M n of n × n complex matrices or the real linear space H n of n × n hermitian matrices. For C ∈ M , its unitary similarity orbit is the set U (C) = {UCU ∗ ; U unitary and its circular unitary similarity orbit is the set V (C) = {μX : μ ∈ F , |μ| = 1, X ∈ U (C)} where F is the scalar field C or R according as M = M n or M = H n . Related to U ( C ) and V ( C ) are the C-numerical range and the C-numerical radius of A ∈ M defined by W c (A) = {tr(AX) : X ∈ U (C)} and r C (A)=max{|z|:z∈W C (A)} , respectively. Let C , D ∈ H n , we study the linear operators T on M satisfying one of the following properties: (I) W D ( T ( A )) = W C ( A ) for all A ∈ M , (II) r D ( T ( A )) = r C ( A ) for all A ∈ M , (III) T ( U (D)) = U(C), (IV) T ( V ( D )) = V ( C ). In particular, we determine the conditions on C and D for the existence of a linear operator T on M satisfying any one of the conditions (I)–(IV), and characterize such an operator if it exists.

We give the best possible bound from below in the Lowener ordering for the Hadamard product A ∘ B in terms of a multiple of A when A is positive semidefinite nonzero and B is positive definite.

Let V be one of the following four real vector spaces: Jn, the n × n real symmetric matrices; Hn, the n × n complex hermitian matrices; M(n, R), the n × n real matrices, and M(n, C), the n × n complex matrices. Suppose T is an R-linear map on V preserving the invertible matrices in the case V = M(n, R) or M(n, C) or preserving the nonsingular balanced inertia class (n even) in the case V = Jn or Hn. If n > 2 and n ≠ 4 or 8 when V = M(n, R), we show that T must be invertible and specify the exact form of T.

When A and B are n × n positive semi-definite matrices, and C is an n × n Hermitian matrix, the validity of a quadratic inequality (x∗Ax)12(x∗Bx)12 ⩾ ¦x∗Cx¦ is shown to be equivalent to the existence of an n × n unitary matrix W such that A12WB12 + B12W∗A12 = 2C. Some related inequalities are also discussed.

For a Hermitian n × n matrix of the form H = P ρQ ρ Q ∗ R of which all the eigenvalues of the s × s submatrix P are greater than all the eigenvalues of the square t × t submatrix R it is proved that the s greater eigenvalues of H are increasing and the remaining t eigenvalues of H are decreasing functions of the absolute value of the complex variable ρ.

Using the resolvent method and the technique of weighted L2-estimates we deduce the non-existence of the singular continuous spectrum of the Dirac operator τ + P(x) on the interval (1, ∞). We assume that the hermitian matrix potential P(x) = P1(x) + P2(x) is divided into a long-range part P 1 (x) = O(¦x¦ −e ), [r ∂ r P 1 (x)] − = O(¦x¦ −e ) and a short-range part P 2 (x) = O(¦x¦ −1 − e ) (¦x¦ → ∞) with local singularities P(x) = O(¦x − a j ¦ −1 ) of Coulomb type for N nuclei a1,…,aN∈ R 3.

This paper consists of two parts. In the first part, we let H be an n × n Hermitian matrix with eigenvalues λ 1 ⩾ ⋯ ⩾ λ n , and let 2 ⩽ k ⩽ n , 1 ⩽ r ⩽ n ⧸2 be fixed with kr ⩽ n . We show that as U varies over all unitary matrices, the eigenvalues η 1 (i) ⩾ ⋯ ⩾ η n−r (i) of the principal submatrices UHU ∗ (1 + (i−1)r,…,ir | 1 + (i−1)r,…,ir) of UHU ∗ , for 1 ⩽ i ⩽ k , independently assume all values permitted by the interlacing inequalities λ j ⩾ η j (i) ⩾ λ j+r , j = 1,…,n − r , (I) if and only if each distinct eigenvalue of H has multiplicity at least rk . Parallel to (I), it is known that the singular values of a submatrix also satisfy certain interlacing inequalities. In the second part, when A is an n × n complex matrix, k and r as above, we give some necessary conditions for the independence of singular values of UAV (1 + ( i − 1) r ,…, ir | 1 + ( i − 1) r ,…, ir ), for 1 ⩽ i ⩽ k , when U and V vary over all unitary matrices. In some cases, the conditions are also proved to be sufficient.

We discuss iterative methods for the solution of the linear system Ax = b , which are based on a single splitting or a multisplitting of A . In order to compare different methods, it is common to compare the spectral radius of the iterative matrix. For M -matrices A and weak regular splittings there exist well-known comparison theorems. Here, we give a comparison theorem for splittings of Hermitian positive definite matrices. Furthermore, we establish a comparison theorem for multisplittings of a Hermitian positive definite matrix.

Let Mn be the algebra of all n × n complex matrices. For 1 ⩽ k ⩽ n, the kth numerical range of A Mn is defined by Wk(A) = (1/k)∑jk=1xj*Axj : x1, …, xk is an orthonormal set in ℂn]. It is known that tr A/n = Wn(A)⊆ Wn−1(A) ⊆ ⋯ ⊆ W1(A). We study the condition on A under which Wm(A) = Wk(A) for some given 1 ⩽ m < k ⩽ n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.

Recently, K. I. Gross and the author [J. Approx. Theory 59:224–246 (1989)] analyzed the total positivity of kernels of the form K(x,y) = p F q (xy) , x,y ϵ R, wherep F q denotes a classical generalized hypergeometric series. There, we related the determinants which define the total positivity of K to the hypergeometric functions of matrix argument, defined on the space of n × n Hermitian matrices. In the first part of this paper, we apply these methods to derive the total positivity properties of the kernels K for more general choices of the parameters in these hypergeometric series. In particular, we prove that if ai > 0 and ki is a positive integer (i = 1,…,p) then K(x,y) = p F q (a 1 ,…,a p ;a 1 + k 1 ,…,a p + k p ; xy) is strictly totally positive on R2. In the second part, we use the theory of entire functions to derive some Polya frequency function properties of the hypergeometric series p F q(x). Some of these latter results suggest a curious duality with the total positivity properties described above. For instance, we prove that if p > 1 and the ai and ki are as above, then there exists a probability density function f on R, such that f is a strict Polya frequency function, and 1/ p F p (a 1 +k 1 ,…,a p +k p ;a 1 ,…,a p ;z= L f(z) , the Laplace transform of f.

The distribution of spectra of scalar and matrix polynomials generated by solutions of Yule-Walker type equations with respect to the real line and the unit circle is investigated. A description of the spectral distribution is given in terms of the inertia of the corresponding hermitian block Hankel or block Toeplitz matrix. These results can be viewed as matrix analogues of M.G. Krein's theorems on polynomials orthogonal on the unit circle.

The Loewner partial order ≧ is defined on the space of Hermitian matrices by A ≧ B if A − B is positive semidefinite. Given a strictly increasing function f: (a, b) → R, we define the partial order ≧f on the set of Hermitian matrices with spectrum contained in (a, b) by A ≧fB if f(A) ≧ f(B). We say that the partial orders ≧ and ≧f are equivalent ona set S of Hermitian matrices if A ≧ B if and only if A ≧fB for all A,B ϵ S. It is clear that if the cone C is commutative, i.e., AB = BA for all A, B ϵ C, then the two partial orders are equivalent. Stepniak conjectured the converse for the function f(t) = t2, and proved it for n ⩽ 3. We provide a counterexample to Stepniak's conjecture for n ≧ 4, and we characterize the convex cones C of positive semidefinite matrices on which ≧ and ≧f are equivalent for a class of functions that includes f(t) = tp, p > 1, and f(t) = et. We introduce the class of strongly monotone matrix functions and prove the following result of independent interest: Let ƒ be a strongly monotone matrix function of order n, and suppose that A and B are n-by-n Hermitian matrices such that A − B is positive semidefinite and that A and B have no common eigenvectors. Then f(A) − f(B) is positive definite. We also show that the functions f(t) = tp, 0 < p < 1, and f(t) = log t are strongly monotone of all orders. We also consider the partial order ≧t2. In this special case it is possible to obtain the results using more elementary techniques. We prove some results about the partial orders ≧tp and ≧exp. We conclude with two open questions.

For square matrices, the relationship is discussed between the notion of Schur complement and the (equivalent) concepts of matricial coupling and equivalence after extension. Special attention is paid to the Hermitian and skew-Hermitian cases.

Let A be a real or complex n × n interval matrix. Then it is shown that the Neumann series Σ ∞ k=0 A k is convergent iff the sequence { A k} converges to the null matrix O , i.e., iff the spectral radius of the real comparison matrix B constructed in [2] is less than one.

In this paper we study the linear Hamiltonian difference system Δy(t) = B(t) y(t + 1) + c(t) z(t) Δz(t) = −A(t) y(t + 1) −B ∗ (t) z(t) , where A(t), C(t) are Hermitian matrices with I − B(t) and C(t) invertible (I is identity). Disconjugacy criteria analogous to those for linear Hamiltonian differential systems are obtained by a discrete Riccati method.

Let A and B be two n × n real symmetric matrices. A theorem of Calabi and Greub-Milnor states that if n ⩾3 and A and B satisfy the condition (uAu′) 2 + (uBu′) 2 ≠ 0 for all nonzero vectors u , then there is a linear combination of A and B that is definite. In this note, the author proves two theorems of the semi-definiteness of a nontrivial linear combination of A and B by replacing the condition ( ∗ ) by another condition. One of these theorems is a generalization of the theorem of Greub-Milnor and Calabi.

This note contains a demonstration of the convexity of the joint range of three Hermitian forms, and, as an immediate corollary, of the convexity of the numerical range of arbitrary sesquilinear forms.

We consider the perturbation properties of the eigensolution of Hermitian matrices. For the matrix entries and the eigenvalues we use the realistic “floating-point” error measure |δa/a|. Recently, Demmel and Veselic considered the same problem for a positive definite matrix H, showing that the floating-point perturbation theory holds with constants depending on the condition number of the matrix A=DHD, where Aii=1 and D is a diagonal scaling. We study the general Hermitian case along the same lines, thus obtaining new classes of well-behaved matrices and matrix pairs. Our theory is applicable to the already known class of scaled diagonally dominant matrices as well as to matrices given by factors—like those in symmetric indefinite decompositions. We also obtain norm estimates for the perturbations of the eigenprojections, and show that some of our techniques extend to non-Hermitian matrices. However, unlike in the positive definite case, we are still unable to describe simply the set of all well-behaved Hermitian matrices.

Finite dimensional indefinite inner product spaces of vector valued rational functions which are (1) invariant under the generalized backward shift and (2) subject to a structural identity, and subspaces and “superspaces” thereof are studied. The theory of these spaces is then applied to deduce a generalization of a pair of rules due to lohvidov for evaluating the inertia of certain subblocks of Hermitian Toeplitz and Hermitian Hankel matrices. The connecting link rests on the identification of a Hermitian matrix as the Gram matrix of a space of vector valued functions of the type considered in the first part of the paper. Corresponding generalizations of another pair of theorems by lohvidov on the rank of certain subblocks of non-Hermitian Teoplitz and non-Hermitian Hankel matrices are also stated, but the proofs will be presented elsewhere.

Let A be an n × n normal matrix, and let 1 ⪕ m . Let α,β ϵ Q m,n , the set of increasing integer sequences of length m chosen from 1, 2,…, n . Suppose α and β have exactly k common entries, denoted by ∥ α ∩ β ∥ = k , and suppose k ⪕ m − 1 . Marcus and Filippenko obtained an upper bound for ∥det A [ α ∥ β ]∥, which depends on k and the moduli of the eigenvalues of A . Using a different approach, we improve their bound. It is also proved that if A is semidefinite hermitian matrix, then max ∥ det U ∗ AU[α∥β]∥ , as U ranges over all n × n unitary matrices and α,β range over all sequences in Q m,n such that ∥ α ∩ β ∥ = k , is a monotonic nondecreasing function of k , confirming a conjecture of Marcus and Moore.

The Kreiss matrix theorem asserts that a family of N × N matrices is L 2 -stable if and only if either a resolvent condition (R) or a Hermitian norm condition (H) is satisfied. We give a direct, considerably shorter proof of the power-boundedness of an N × N matrix satisfying (R), sharpening former results by showing that power- boundedness depends, at most, linearly on the dimension N . We also show that L 2 -stability is characterized by an H-condition employing a general H-numerical radius instead of the usual H-norm, thus generalizing a sufficient stability criterion, due to Lax and Wendroff.

We present general recurrences for the Pade table that allow us to skip ill- conditioned Pade approximants while we proceed along a row of the table. In conjunction with a certain inversion formula for Toeplitz matrices, these recurrences form the basis for fast algorithms for solving non-Hermitian Toeplitz systems. Under the assumption that the lookahead step size (i.e., the number of successive skipped approximants) remains bounded, we give both O ( N 2 ) and O ( N log 2 N ) algorithms which are (presumably) weakly stable. With little additional work, still in O ( N 2 ) operations, we can also obtain a decomposition of the Toeplitz matrix T according to TR = LD , where R is upper triangular, L is unit lower triangular, and D is block-diagonal. The relation to continued fractions is also discussed.

The author observes that two Hermitian forms have the same largest eigenvalue. A large sieve result of Roth-Bombieri type and Selberg's upper bound sieve with a Montgomery type error term are derived.

It is shown that the ratio of the area of the convex hull of the fields of values of the (n−1)-by-(n−1) principal submatrices of an n-by-n matrix A to the area of the field of values of A is bounded below by a function of n which approaches 1 as n approaches ∞. Since this convex hull is necessarily contained in the field of values of A, an interpretation is that, asymptotically in the dimension, the field of any given matrix is “filled up” by the fields of the submatrices (collectively). Some new inequalities for the eigenvalues of principal submatrices of hermitian matrices, which are not implied by interlacing, are employed.

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem ∂u∂t + ∑j = 1n Aj∂u∂xj = 0, u(0, x) = ƒ(x). Here the Aj's are constant, k × k Hermitian matrices, x = (x1,…, xn), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit EM(ƒ) as ¦ t ¦ → ∞, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ƒ, which is invariant under the solution group U0(t) and such that U0(t)ƒ = 0 for ¦ x ¦ ⩽ a ¦ t ¦ − R, a and R depending on ƒ and that the local energy of nonstatic solutions decays as ¦ t ¦ → ∞. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation E(x) ∂u∂t + ∑j = 1n Aj∂u∂xj = 0, where ¦ E(x) − I ¦ = O(¦ x ¦−1 − ϵ) at infinity.

Asymptotic growth rates are obtained for the eigenvalues of a partitioned Hermitian matrix having positive definite and negative definite complementary principal submatrices, when the nonprincipal submatrix grows indefinitely large.

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 aij = 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.