Eigenvalue inequalities for principal submatrices

For a polynomial with real roots, inequalities between those roots and the roots of the derivative are demonstrated and translated into eigenvalue inequalities for a hermitian matrix and its submatrices. For example, given an n-by-n positive definite hermitian matrix with maximum eigenvalue λ, these inequalities imply that some principal submatrix has an eigenvalue exceeding $\text{[(}\text{n?1)}\text{n}\text{]λ}$.     

Eigenvalue inequalities and equalities

We consider cases of equality in three basic inequalities for eigenvalues of Hermitian matrices: Cauchy's interlacing inequalities for principal submatrices, Weyl's inequalities for sums, and the residual theorem. Several applications generalize and sharpen known results for eigenvalues of irreducible tridiagonal Hermitian matrices.     

Singular values of matrix exponentials

The singular values of a matrix and those of its exponential are related via multiplicative majorization. Matrices giving some equalities in the majorization are characterized. As an application, a scalar inequality for the exponential function is generalized to a matrix-valued inequality and the case of equality is examined.     

Eigenvalue inequalities for convex and log-convex functions

We give a matrix version of the scalar inequality f(a + b) ? f(a) + f(b) for positive concave functions f on [0, ∞). We show that Choi’s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences. Some inequalities for log-convex functions are presented and a new arithmetic-geometric mean inequality for positive matrices is given. We also point out a simple proof of the Bhatia-Kittaneh arithmetic-geometric mean inequality.     

Computing multiple integrals involving matrix exponentials

In this paper, a generalization of a formula proposed by Van Loan [Computing integrals involving the matrix exponential, IEEE Trans. Automat. Control 23 (1978) 395–404] for the computation of multiple integrals of exponential matrices is introduced. In this way, the numerical evaluation of such integrals is reduced to the use of a conventional algorithm to compute matrix exponentials. The formula is applied for evaluating some kinds of integrals that frequently emerge in a number classical mathematical subjects in the framework of differential equations, numerical methods and control engineering applications.     

Eigenvalue inequalities in an embeddable factor

We provide a characterization of the possible eigenvalues of the sum of two selfadjoint elements of a II factor which can be embedded in the ultrapower of the hyperfinite II factor.

     

Local Remez-type inequalities for exponentials of a potential on a piecewise analytic ARC

Based on techniques from potential theory and geometric function theory, we give a new proof of the pointwise Remez-type inequality for exponentials of logarithmic potentials on the unit interval [−1, 1]. This inequality was first proved by Erdélyi, Li and Saff [8, Theorem 2.2.]. We generalize our result to the case of a piecewise analytic arc.     

Eigenvalue inequalities for the buckling problem of the drifting Laplacian

In this paper, we study the buckling problem of the drifting Laplacian on bounded domains in a complete Riemannian manifold with nonnegative ∞-dimensional Bakry–Émery Ricci curvature. According to the property of the manifold, we obtain a family of trial functions. By making use of these trial functions, we derive a universal inequality of eigenvalues, which is independent of the domains.     

Tail behavior of random products and stochastic exponentials

In this paper we study the distributional tail behavior of the solution to a linear stochastic differential equation driven by infinite variance α$\alpha$-stable Lévy motion. We show that the solution is regularly varying with index α$\alpha$. An important step in the proof is the study of a Poisson number of products of independent random variables with regularly varying tail. The study of these products merits its own interest because it involves interesting saddle-point approximation techniques.     

Eigenvalue inequalities associated with the cartesian decomposition

Let T=A+iB where AB are Hermitian matrices. We obtain several inequalities relating the l_p 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.     

Trace inequalities for products of matrices

In this short paper, we study some trace inequalities of the products of the matrices and the power of matrices by the use of elementary calculations.     

Eigenvalue problems for nonlinear elliptic variational inequalities on a cone

Eigenvalue problems for variational inequalities on a closed convex cone C in a real Banach space V, of the form 〈g′(v) ? λh′(v), w ? v〉 ? 0 for all w in C, are considered with the normalization g(v) = r, where g and h are real-valued C¹ functions on V. Theorems are proved on the existence of solutions λ(r) and v(r), and on their dependence upon the normalization constant r > 0. In particular, the relation, as r → 0, of λ(r), v(r) to solutions of the analogous problem with g″(0) and h″(0) in place of g′ and h′, is discussed. The theorems are applied to elliptic inequalities for Euler-Lagrange operators corresponding to multiple integral functionals on closed subspaces of Sobolev spaces, and to the inequality arising from the von Karman equations for the buckling of a thin elastic plate which is constrained to buckle in only one direction.