Off-diagonal submatrices of a Hermitian matrix

A necessary and sufficient condition is given to a complex matrix to be an off-diagonal block of an Hermitian matrix with prescribed eigenvalues (in terms of the eigenvalues of and singular values of ). The proof depends on some recent breakthroughs in the study of spectral inequalities on the sum of Hermitian matrices by Klyachko and Fulton. Some interesting geometrical properties of the set of all such matrices are derived from the main result. These results improve earlier ones that only give partial information for the set .

     

Almost orthogonal operators on the bitorus

The operators S _p f (x, y), for the sum of which we prove an L ²-estimate, act as a kind of Fourier coefficients on one variable and a kind of truncated Hilbert transforms with a phase N(x, y) on the other variable. This result is an extension to two-dimensions of an argument of almost orthogonality in Fefferman’s proof of a.e. convergence of Fourier series, under the basic assumption N(x, y) “mainly” a function of y and the additional assumption N(x, y) non-decreasing in x, for every y fixed.     

Almost orthogonal operators on the bitorus II

We prove that the operator ${Tf(x,y)=\int^\pi_{-\pi}\int_{|x^{\prime}|<|y^{\prime}|} \frac{e^{iN(x,y) x^{\prime}}}{x^{\prime}}\frac{e^{iN(x,y) y^{\prime}}}{y^{\prime}}f(x-x^{\prime}, y-y^{\prime}) dx^{\prime} dy^{\prime}}$ , with ${x,y \in[0,2\pi]}$ and where the cut off ${|x^{\prime}|<|y^{\prime}|}$ is performed in a smooth and dyadic way, is bounded from L ² to weak- ${L^{2-\epsilon}}$ , any ${\epsilon > 0 }$ , under the basic assumption that the real-valued measurable function N(x, y) is “mainly” a function of y and the additional assumption that N(x, y) is non-decreasing in x, for every y fixed. This is an extension to 2D of C. Fefferman’s proof of a.e. convergence of Fourier series of L ² functions.     

Minimal polynomial and the rank of principal submatrices of a matrix

The following theorem is proved: If r is the degree of the minimal polynomial of a matrix A then there exists a principal submatrix of A with order r and rank at least r-1.     

The inertia of a Hermitian matrix having prescribed complementary principal submatrices

For i=1,2 let H_i be a given n_i×n_i Hermitian matrix. We characterize the set of inertias

$\text{In}\text{H}{}_1\text{X}\text{X}{}^1\text{H}{}_2\text{:X}\text{is}\text{n}{}_1\text{×n}{}_2$

in terms of In(H₁) and In(H₂).     

The eigenvalue spreads of a hermitian matrix and its principal submatrices

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.     

The eigenvalue spreads of a hermitian matrix and its principal submatrices

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.     

Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA *=B with applications

Suppose that AXA ^*=B is a consistent matrix equation and partition its Hermitian solution X ^*=X into a 2-by-2 block form. In this paper, we give some formulas for the maximal and minimal ranks of the submatrices in an Hermitian solution X to AXA ^*=B. From these formulas we derive necessary and sufficient conditions for the submatrices to be zero or to be unique, respectively. As applications, we give some properties of Hermitian generalized inverses for an Hermitian matrix.     

Random orthogonal matrix simulation

This paper introduces a method for simulating multivariate samples that have exact means, covariances, skewness and kurtosis. We introduce a new class of rectangular orthogonal matrix which is fundamental to the methodology and we call these matrices L matrices. They may be deterministic, parametric or data specific in nature. The target moments determine the L matrix then infinitely many random samples with the same exact moments may be generated by multiplying the L matrix by arbitrary random orthogonal matrices. This methodology is thus termed “ROM simulation”. Considering certain elementary types of random orthogonal matrices we demonstrate that they generate samples with different characteristics. ROM simulation has applications to many problems that are resolved using standard Monte Carlo methods. But no parametric assumptions are required (unless parametric L matrices are used) so there is no sampling error caused by the discrete approximation of a continuous distribution, which is a major source of error in standard Monte Carlo simulations. For illustration, we apply ROM simulation to determine the value-at-risk of a stock portfolio.     

The possible inertias for a Hermitian matrix and its principal submatrices

Definition: A Hermitian matrix H is a Hermitian extension of a given set of Hermitian matrices {H_ii, i = 1,…,m} if these {H_ii} are the block diagonals of H. Let (π_i,v_i,δ_i) = InH_ii, the inertia of each H_ii. Special case: Given Hermitian matrices {H_ii, i=1,…,m} and given nonnegative integers π, v, and δ such that π+v+δ=Σ(π_i+v_i+δ_i); then a Hermitian extension H exists such that Ker H⊃⊕KerH_ii and InH=(π, v, δ) if and only if δ⩾Σδ_i and π⩾max π_i and v⩾maxv_i. We also present a simple extension theorem for the general case (KerH ⊅ ⊕ Ker H_ii).     

A method of matrix inverse triangular decomposition based on contiguous principal submatrices

An algorithm is presented which performs the triangular decomposition of the inverse of a given matrix. The method is applicable to any matrix all contiguous principal submatrices of which are nonsingular. The algorithm is particularly efficient when the matrix has certain partial symmetries exhibited by the Toeplitz structure.     

Discrete orthogonal matrix polynomials

At the present time, the theory of orthogonal matrix polynomials is an active area of mathematics and exhibits a promising future. However, the discrete case has been completely forgotten. In this note we introduce the notion of discrete orthogonal matrix polynomials, and show some algebraic properties. In particular, we study a matrix version of the usual Meixner polynomials.     

Almost reduction and perturbation of matrix cocycles

In this note, we show that if all Lyapunov exponents of a matrix cocycle vanish, then it can be perturbed to become cohomologous to a cocycle taking values in the orthogonal group. This extends a result of Avila, Bochi and Damanik to general base dynamics and arbitrary dimension. We actually prove a fibered version of this result, and apply it to study the existence of dominated splittings into conformal subbundles for general matrix cocycles.     

Forbidden submatrices

We consider a m × n (0, 1)-matrix A, no repeated columns, which has no k × l sumatrix F. We may deduce bounds on n, polynomial in m, depending on F. The best general bound is O(m^2k−1). We improve this and provide best possible bounds for k × 1 F's and certain k × 2 F's. In the case that all columns of F are the same, good bounds are obtained which are best possible for l = 2 and some other cases. Good bounds for 1 × l F's are provided, namely n ⩽ (l−1)m + 1, which are shown to be best possible for F = [1010...10]. The paper finishes with a study of the 14 different 3 × 2 possibilities for F, solving all but 3.     

Operator norms of submatrices

Translated from Matematicheskie Zametki, Vol. 45, No. 3, pp. 94–100, March, 1989.     

On orthogonal systems of matrix algebras

In this work it is shown that certain interesting types of orthogonal system of subalgebras (whose existence cannot be ruled out by the trivial necessary conditions) cannot exist. In particular, it is proved that there is no orthogonal decomposition of M_n(C)⊗M_n(C)≡M_n²(C) into a number of maximal abelian subalgebras and factors isomorphic to M_n(C) in which the number of factors would be 1 or 3.In addition, some new tools are introduced, too: for example, a quantity c(A,B), which measures “how close” the subalgebras A,B⊂M_n(C) are to being orthogonal. It is shown that in the main cases of interest, c(A^′,B^′) - where A^′ and B^′ are the commutants of A and B, respectively - can be determined by c(A,B) and the dimensions of A and B. The corresponding formula is used to find some further obstructions regarding orthogonal systems.