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Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, and para-Hermitian rational matrices arise in many applications. Linearizations of rational...

Using the method of seeds and branch duplication, it is shown that for every tree of diameter , there is an Hermitian matrix with as few as distinct eigenvalues (a known lower bound). For diameter 7, some trees require 8 distinct eigenvalues, but no more; the seeds for which 7 and 8 are the worst case are classified. For trees of diameter , it is shown that, in general, the minimum number of distinct eigenvalues is bounded by a function of . Many trees of high diameter permit as few of distinct eigenvalues as the diameter and a conjecture is made that all linear trees are of this type. Several other specific, related observations are made.

In matrix theory, C.-K. Li and R.-C. Li gave the best bound of the difference between eigenvalues of a real/complex Hermitian matrix and the matrix after removing off-diagonal blocks. In this paper, we extend this result to the setting of simple Euclidean Jordan Algebras.

Letbe two Hermitian matrices. We propose new perturbation bounds on the differences between the eigenvalues of and by the bounds of the eigenvector components. These results extend those of Nakatsukasa.

There are numerous min–max principles about the eigenvalues of a Hermitian matrix. The most general ones are Wielandt’s min–max principles which include the Courant–Fischer min–max principles and the trace minimization principles as special cases. In this paper, various extensions of Wielandt’s principles are obtained for a positive semi-definite pencil by which we mean that A and B are Hermitian and there is a real number such that is positive semi-definite.

In this paper, we introduce a new class of upper triangular nilpotent matrices, etermine the eigenvalues of the Hermitian part of rotating the type of nilpotent matrices and relate the numerical range of that typical type of matrices to the orbit of a point mass under a central force.

We discuss some applications of generalized interlacing inequalities of Ky Fan to the study of (a) some classical matrix inequalities and (b) matrix problems in quantum information science.

Let be a Hermitian matrix. It is known that the vector of diagonal elements of H, diag(H), is majorized by the vector of the eigenvalues of H, λ(H), and that this majorization can be extended to the eigenvalues of diagonal blocks of H. Reverse majorization results for the eigenvalues are our goal. Under the additional assumptions that H is positive semidefinite and the block K is Hermitian, the main result of this article provides a reverse majorization inequality for the eigenvalues. This results in the following majorization inequalities when combined with known majorization inequalites on the left

A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2 × 2 and 3 × 3, as well as k × k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold.

We investigate the relation between perturbing the i-th diagonal entry of A ∈ ℳ n (𝔽) and extracting the principal submatrix A(i) from A with respect to the possible changes in multiplicity of a given eigenvalue. A complete description is given and used to both generalize and improve prior work about Hermitian matrices whose graph is a given tree.

Given commuting families of Hermitian matrices {A 1, … , A k } and {B 1, … , B k }, conditions for the existence of a completely positive map Φ, such that Φ(A j ) = B j for j = 1, … , k, are studied. Additional properties such as unital or/and trace preserving on the map Φ are also considered. Connections of the study to dilation theory, matrix inequalities, unitary orbits and quantum information science are mentioned.

We establish some identities for the characteristic polynomial of Hermitian matrices whose graph is a cycle. We use the paper C.M da Fonesca: Interlacing properties for Hermitian matrices whose graph is a given tree, SIAM J. Matrix Anal. Appl. 27 (2005) pp. 130–141 and we extend some interlacing results obtained in this article to graphs with a cut edge. For some cases, we give a graphical interpretation of the results.

Let A be an n by n real symmetric matrix, and let α be a non-empty subset of {1, …, n}. The principal submatrix of A obtained by removing rows and columns indexed by α is denoted by A(α). If the nullity of A(α) is |α| more than that of A, then α is called a P-set of A, and the indices in α are called P-vertices of A. This article concerns P-vertices and P-sets of non-singular acyclic matrices A when the graph T of A is a path or a star. It was shown in [C.R. Johnson and B.D. Sutton, Hermitian matrices, eigenvalue multiplicities, and eigenvector components, SIAM J. Matrix Anal. Appl. 26 (2004), pp. 390–399] that each singular acyclic matrix of order n has at most n − 2 P-vertices. In this article we show that this does not hold for non-singular acyclic matrices by constructing non-singular acyclic matrices, whose graphs are T, having n − 1 (or n) P-vertices (these matrices also achieve the maximum size of a P-set over non-singular acyclic matrices whose graphs are T). In addition, it is shown that if n is ...

Suppose that the eigenvalues of an Hermitian matrix A whose graph is a tree T are known, as well as the eigenvalues of the principal submatrix of A corresponding to a certain branch of T. A method for constructing a larger tree T ', in which the branch is ‘`duplicated’', and an Hermitian matrix A′ whose graph is T ' is described. The eigenvalues of A' are all of those of A, together with those corresponding to the branch, including multiplicities. This idea is applied (1) to give a solution to the inverse eigenvalue problem for stars, (2) to prove that the known diameter lower bound, for the minimum number of distinct eigenvalues among Hermitian matrices with a given graph, is best possible for trees of bounded diameter, and (3) to increase the list of trees for which all possible lists for the possible spectra are know. A generalization of the basic branch duplication method is presented.

Suppose k 1

Normal matrices with respect to indefinite inner products are studied using the additive decomposition into selfadjoint and skewadjoint parts. In particular, several structural properties of indecomposable normal matrices are obtained. These properties are used to describe classes of matrices that are logarithms of selfadjoint or normal matrices. In turn, we use logarithms of normal matrices to study polar decompositions with respect to indefinite inner products. It is proved, in particular, that every normal matrix with respect to an indefinite inner product defined by an invertible Hermitian matrix having at most two negative (or at most two positive) eigenvalues, admits a polar decomposition. Previously known descriptions of indecomposable normals in indefinite inner products with at most two negative eigenvalues play a key role in the proof. Both real and complex cases are considered.

If A and B are n-by-n, positive semidefinite Herimitian matrices, then for any φ≠α⊆{1,2,… n}. A certain converse is given, as well as analogs for the product of several m-matrices and totally nonnegative matrices. However, analogs for λmax (other than the case of totally nonnegative matrices) and for singular values of general matrices, etc., fail.

In this note we show that a well known relationship between the Lowner partial ordering and the space preordering within the class of non-negative definite matrices remains true for Hermitian matrices with the same number of negative eigenvalues. Some consequences in connection with other partial orderings are discussed.

In this note, we prove the following theorem. Given 2n real numbers x 1≥β1≥x,≥≥β2≥ [cddot] x n ≥β n ≥0 and a directed tree T on n+1 vertices with a distinguished vertex which is a source. then there exists an n+1 x n real matrix A that satisfies. 1. The directed graph of A is T (where the first row of A corresponds to the distinguished source vertex); 2. The singular values of A are α1, …, α n 3. The singular values of the n × n matrix A 0 obtained by deleting the top row of A are β1, …, β n We adapt out method to prove a similar result first found by A. L. Duarte for eigenvalue interlacing of Hermitian matrices. Some partial results of this type for invariant factor interlacing of integral matrices are given.

For 1≤m≤n, the mth decomposable numerical range of an nxn complex matrix A is defined and denoted by In this note, we determine the conditions on as essentially Hermitian matrix A(i.e.A is normal with colliner eigenvalues) such that is (i) convex, or (ii) simply connected. For a general A, we obtain a sufficient condition for to be starshaped. These answer some questions raised by Bebiano and the second author on the geometrical properties of the decomposable numerical range.

We prove some inequalities involving the eigenvalues of an nxn Hermitian matrix and the eigenvalues of the (n−1)x(n−1) principal submatrices. We apply this inequality to generalize a known result on the numerical range to the lth numerical range. The method used yields an elegant proof of the converse to the interlacing theorem, which we include. A counterexample to the quardratic spread inequality conjectured by R. C. Thompson is also given.

We investigate a natural generalization of the problem of the description of the eigenvalues of the sum of two Hermitian matrices both of whose eigenvalues are known. We describe more generally the convolution of the invariant probability measures supported on any two adjoint orbits of a compact Lie group. Our techniques utilize the convexity results of Guillemin and Sternberg and Kirwan on the one hand, and the character formulae of Weyl and Kirillov on the other. Applications to representation theory are discussed.

Let λ1…λ n be distinct complex numbers and let λ i1…λ in−1 ,i= 1,…n, be complex numbers, not necessarily distinct. We show that the problem of determining when a normal n × n matrix A exists satisfying σ(A) ={λ1…λ n } and σ(Ai )={λ i1…λ in−1 },i=1,…,n where Ai is the (n−1) × (n−1) principal submatrix obtained from A by deleting its ith row and column, is equivalent to determining whether a certain n×n matrix constructed from the λ i ,λ ij is unistochastic. This result, coupled with the characterization of unistochastic 3 × 3 matrices in [2], allows a characterization of the possible eigenvalues of the three 2 × 2 principal submatrices of a normal, Hermitian or real symmetric 3 × 3 matrix. Further results obtained involve the possible eigenvalues of a single (n−1) × (n−1) principal submatrix of a normal n × n matrix and the possible eigenvalues of a pair of (n−1) × (n−1) principal submatrices of a normal, Hermitian or real symmetric n × n matrix. Some of these results are extended to the case where the λ i...

Bounds are given for eigenvalues of hermitian completions of partial hermitian matrices, and for singular values of completions of partial block triangular matrices.

Inequalities are proved connecting the eigenvalue spread of a Hermitian matix to the eigenvalue spreads of its collection of principal submatrices. An application is made to the numerical range of general matrices.

A norm N(·) on Hn the space of all n × n hermitian matrices, is unitary similarity invariant if N(UAU* ) = N(A) for all hermitian matrices A and all unitary matrices U. We characterize those linear isometries on Hn with respect to a unitary similarity invariant norm. A related result concerning the characterization of those linear operators on Hn which preserve the set of matrices whose vectors of eigenvalues belong to a certain compact subset of is also obtained.

Let a positive definite Hermitian matrix HϵMn (C) be decomposed as H=A + iB, with A, B ϵ Mnm (R). We give two new proofs of the inequality det H ≤ det A (with equality iff B = 0. each of which vields something futher. One exhibits majorization between the eigenvalues of A and H the other allows proof of the permanental analog per H≥per A.

Let A be a bounded linear operator in a quaternionic Hilberi space (H,(·,·)). The numerical range of A is defined to be the set W (A)={( Au, u): u e H, (u,u) = 1}. Quite different from the complex caseW (A) may not be convex. In this note the author proves that W (A) is convex if and only if R ∩ W (A) = {Req : q e W(A)}. where R is the real field and Re q denotes the real part of the quaternion q. For a normal operator A in a finite dimensional space H, the author gives a characterization on the convexity of W(A) in terms of ihe eigenvalues of A and also proves that the generalized numerical range of A is convex if and only if A is Hermitian.

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.

The main result of this paper is the following Let A and B be n×n hermitian matrices with eigenvalues respectively, ordered so that and let M1 be any k×k principal submatrix of . Necessary and sufficient conditions for equality are given.

A partial Hermitian matrix is one in which some entries are specified and others are considered to be free (complex) variables. Assuming the undirected graph of the specified entries is chordal, it is shown that, with certain mild restrictions, a partial Hermitian matrix may be completed to a Hermitian matrix with any inertia allowed by the specified principal submatrices through the interlacing inequalities. This generalizes earlier work dealing with the existence of positive definite completions, and. as before, the chordality assumption is, in general, necessary. Further related observations dealing with Toeplitz completions and the minimum eigenvalues of completions are also made, and these raise additional questions.

We give some steps towards a unified theory of Courant-Fischer minimax-type formulas and Cauchy interlacing-type inequalities that have been obtained for the eigenvalues of Hermitian matrices, for singular values of complex matrices, and for invariant factors of integral matrices We also unify and extend work on eigenvalues, singular values, and invariant factors of pairs of matrices and their sum or product

Let T=A+iB where A B are Hermitian matrices. We obtain several inequalities relating the lp distance between the eigenvalues of A and those of iB with the Schatten p-norm of T. The majorization results which lead to these inequalities are also used to get simple proofs of some known lower and upper bounds for the determinant of T.

We give a new short proof using properties of the field of values to show that a) a complex matrix with only real eigenvalues is either hermitian or has indefinite imaginary part, and b) one with only purely imaginary eigenvalues is either skew-hermitian or has indefinite real part, while c) one whose eigenvalues all have absolute value 1 is either unitary or has indefinite polar defect I—TT* . Conversely, every skewsymmetric matrix is the skewsymmetric part of some real matrix that is similar to a real diagonal matrix. The corresponding result for complex matrices is found to be false.

Moyls and Marcus [4] showed that for n⩽4,n×n an complex matrix A is normal if and only if the numerical range of A is the convex hull of the eigenvalues of A. When n⩾5, there exist matrices which are not normal, but such that the numerical range is still the convex hull of the eigenvalues. Two alternative proofs of this fact are given. One proof uses the known structure of the numerical range of a 2×2 matrix. The other relies on a theorem of Motzkin and Taussky stating that a pair of Hermitian matrices with property L must commute.

Let Σ(F) be the class of hermitian positive definite elements of Mn (F), where F is either R, the real, or C, the complex field, and let For j ⩾ 0 and k ⩾ 1, all set products of the form: are determined for integers j k. This completes earlier work of Ballantine and Taussky which determined for integers j ⩾ 0. Also inequalities for the eigenvalues of are given in terms of conjunctive invariants of S T ∈ Π (C). Finally some conditions are presented which insure for certain pairs A B ∈ Mn (C) that the product AB is in Π(C).

A majorization result is proved which involves the singular values of a matrix Aand those of are arbitrary matrices of the same order. The result generalizes previous majorization results concerning eigenvalues of hermitian matrices

Let V be an n-dirneasional unitary space with inner product (x y) and let A: V → V be an hermitian operator with eigenvalues λ1 ⩾⋯ ⩾ λn. For each m1 ⩽ m ⩽ n let be the mth exterior space over Vand let G m denote the Grassman manifold, i.e., all decomposable elements of of norm 1. The map C m(A): is the induced compound of A. This paper is concerned with the structure of the set consisting of all complex numbers vary over all orthonormal pairs in Gm . The first result states that is a closed disc centered at the origin. If λ1 …, λn are the eigenvalues of A then the maximum and minimum eigenvalues of Cm (A) are respectively λα. and λβ where both λα = λα1… λαmand λβ = λβ1… λβmare products taken m at a time of the λi i = 1,…,n. The radius of is always at most then has radius ρ iff α and β agree in m− 1 places. For hermitian matrices A this implies that the values of a nonprincipal subdeterminant of U * AU, as U runs over unitary matrices, fills up a disc whose radius depends on the position of the subdeterminant.

The structure of a unital linear map on hermitian matrices with the property that it preserves the set of invertible hermitian matrices with fixed indefinite inertia is examined. It turns out that such a map is either a unitary similarity or a unitary similarity followed by a transposition (the case when the fixed inertia has equal number of positive and negative eigenvalues is excluded).

It is our purpose to compute the maximum value of the modulus of the determinant of an m×m nonprincipal submatrix of an n×n hermitian (or real symmetric) matrix A, in terms of m, the eigenvalues of A, and the cardinality k of the set of common row and column indices of this submatrix.