Let $A$ and $B$ be $n\,\times \,n$ complex Hermitian (or real symmetric) matrices with eigenvalues ${{a}_{1}}\,\ge \,\cdots \,\ge \,{{a}_{n}}$ and ${{b}_{1}}\,\ge \,\cdots \,\ge \,{{b}_{n}}$. All possible inertia values, ranks, and multiple eigenvalues of $A\,+\,B$ are determined. Extension of the results to the sum of $k$ matrices with $k\,>\,2$ and connections of the results to other subjects such as algebraic combinatorics are also discussed.

The length of every pair {A,B} of 6×6 complex matrices is shown to be at most 10, that is, the words in A,B of length at most 10, including the empty word, span the unital algebra generated by A,B. This supports the conjecture that the length of every pair of n×n complex matrices is at most 2n−2, known to be true for n<6.

For two given projections p and q in a C*-algebra, we investigate how to express the Drazin inverses of the product pq and the difference p−q, and give applications. As a special case, we obtain the results of [C. Y. Deng, ‘The Drazin inverses of products and differences of orthogonal projections’, J. Math. Anal. Appl.335 (2007) 64–71], with considerably simpler proofs.

We study the matrix equation C(BXC)†B=X†, where X† denotes the Moore–Penrose inverse. We derive conditions for the consistency of the equation and express all its solutions using singular vectors of B and C. Applications to compliance matrices in molecular dynamics, to mixed reverse-order laws of generalized inverses and to weighted Moore–Penrose inverses are given.

We investigate the perturbation of the palindromic eigenvalue problem for the matrix quadratic with A0,A1∈𝒞n×n and (where or H). The perturbation of eigenvalues in the context of general matrix polynomials, palindromic pencils, (semi-Schur) anti-triangular canonical forms and differentiation is discussed.

In this paper we derive perturbation theorems for the LU and QR factors. Moreover, bounds for κL(A)/κL′(A) and κU(A)/κ′U(A) are given for the LU factorization of a nonsingular matrix. By applying pivoting strategies in the LU factorization, estimates for κL(PAQ)/κL′(PAQ) and κU(PAQ)/κ′U(PAQ) are also obtained.

The structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps f satisfying for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that Γ(f(A))=Γ(A) for various functions Γ including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonné on linear maps preserving the determinant functions and linear maps preserving the set of singular matrices, respectively.

In this article we consider Re-nnd solutions of the equation AXB=C with respect to X, where A,B,C are given matrices. We give necessary and sufficient conditions for the existence of Re-nnd solutions and present a general form of such solutions. As a special case when A=I we obtain the results from a paper of Groß (‘Explicit solutions to the matrix inverse problem AX=B’, Linear Algebra Appl.289 (1999), 131–134).

By means of a symbolic calculus for finding solutions of difference equations, we derive explicit eigenvalues, eigenvectors and inverses for tridiagonal Toeplitz matrices with four perturbed corners.

In this paper we establish the definition of the generalized inverse A(2)T, S which is a {2} inverse of a matrix A with prescribed image T and kernel s over an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverse and some explicit expressions for of a matrix A over an associative ring, which reduce to the group inverse or {1} inverses. In addition, we show that for an arbitrary matrix A over an associative ring, the Drazin inverse Ad, the group inverse Ag and the Moore-Penrose inverse . if they exist, are all the generalized inverse A(2)T, S.

Let V be an n–dimensional inner product space over , let H be a subgroup of the symmetric group on {l,…, m}, and let x: H → be an irreducible character. Denote by (H) the symmetry class of tensors over V associated with H and x. Let K (T) ∈ End((H)) be the operator induced by T ∈ End(V), and let DK(T) be the derivation operator of T. The decomposable numerical range W*(DK(T)) of DK(T) is a subset of the classical numerical range W(DK(T)) of DK(T). It is shown that there is a closed star-shaped subset of complex numbers such that

⊆ W*(DK(T)) ⊆ W(DK(T)) = con

where conv denotes the convex hull of . In many cases, the set is convex, and thus the set inclusions are actually equalities. Some consequences of the results and related topics are discussed.

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

Let Mn, be the algebra of all n × n matrices over a field F, where n ≧ 2. Let S be a subset of Mn containing all rank one matrices. We study mappings φ: S → Mn, such that F(φ (A)φ (B)) = F(A B) for various families of functions F including all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form A ↦ μ(A)S(σ (aij))S-1 for all A= (aij) ∈ S for some invertible S ∈ Mn, field monomorphism σ of F, and an F*-valued mapping μ defined on S. For real matrices, σ is often the identity map; for complex matrices, σ is often the identity map or the conjugation map: z ↦ z. A key idea in our study is reducing the problem to the special case when F:Mn → {0, 1} is defined by F(X) = 0, if X = 0, and F(X) = 1 otherwise. In such a case, one needs to characterize φ: S → Mn such that φ(A) φ (B) = 0 if and only if AB = 0. We show that such a map has the standard form described above on rank one matrices in S.

Let aπ denote the spectral idempotent of a generalized Drazin invertible element a of a ring. We characterize elements b such that 1 − (bπ − aπ)2 is invertible. We also apply this result in rings with involution to obtain a characterization of the perturbation of EP elements. In Banach algebras we obtain a characterization in terms of matrix representations and derive error bounds for the perturbation of the Drazin Inverse. This work extends recent results for matrices given by the same authors to the setting of rings and Banach algebras. Finally, we characterize generalized Drazin invertible operators A, B ∈ (X) such that pr(Bπ) = pr(Aπ + S), where pr is the natural homomorphism of (X) onto the Calkin algebra and S ∈(X) is given.

2000 Mathematics subject classification: primary 16A32, 16A28, 15A09.

We develop several iterative methods for computing generalized inverses using both first and second order optimization methods in C*-algebras. Known steepest descent iterative methods are generalized in C*-algebras. We introduce second order methods based on the minimization of the norms ‖Ax − b‖2 and ‖x‖2 by means of the known second order unconstrained minimization methods. We give several examples which illustrate our theory.

As an attempt to understand linear isometries between C*-algebras without the surjectivity assumption, we study linear isometries between matrix algebras. Denote by Mm the algebra of m × m complex matrices. If k ≥ n and φ: Mn → Mk has the form X ↦ U[X ⊕ f(X)] V or X ↦ U[X1 ⊕ f(X)]V for some unitary U, V ∈ Mk and contractive linear map f: Mn → Mk, then ║φ(X)║ = ║X║ for all X ∈ Mn. We prove that the converse is true if k ≤ 2n - 1, and the converse may fail if k ≥ 2n. Related results and questions involving positive linear maps and the numerical range are discussed.

An optimal coupling is a bivariate distribution with specified marginals achieving maximal correlation. We show that optimal couplings are totally positive and, in fact, satisfy a strictly stronger condition we call the nonintersection property. For discrete distributions we illustrate the equivalence between optimal coupling and a certain transportation problem. Specifically, the optimal solutions of greedily-solvable transportation problems are totally positive, and even nonintersecting, through a rearrangement of matrix entries that results in a Monge sequence. In coupling continuous random variables or random vectors, we exploit a characterization of optimal couplings in terms of subgradients of a closed convex function to establish a generalization of the nonintersection property. We argue that nonintersection is not only stronger than total positivity, it is the more natural concept for the singular distributions that arise in coupling continuous random variables.

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.

We improve on previous finite time estimates for the simulated annealing algorithm which were obtained from a Cheeger-like approach. Our approach is based on a Poincaré inequality.

Additive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if Ad denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B)d is computed in terms of Ad and Bd. Thus, recent results of Hartwig, Wang and Wei (Linear Algebra Appl. 322 (2001), 207–217) are extended to infinite dimensional settings with simplified proofs.