On the semi-definiteness of the real pencil of two Hermitian matrices

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

$\text{(uAu′)}{}^2\text{+ (uBu′)}{}^2\text{≠ 0}$

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 (¹) by another condition. One of these theorems is a generalization of the theorem of Greub-Milnor and Calabi.     

Almost commuting matrices

It is shown that if A and B are n × n complex matrices with $\text{A = A}{}^1\text{and}\text{∥AB ? BA∥</}\text{2?}{}^2\text{(n ? 1)}$, then there exist n × n matrices A′ and B′ with $\text{A′ = A′}{}^1\text{such that}\text{A′B′ = B′A′}\text{and}\text{∥A ? A′∥? ?, ∥B ? B′∥? ?}$.     

Structure of quadratic inequalities

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 $\text{(x}{}^1\text{Ax)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(x}{}^1\text{Bx)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{? ¦x}{}^1\text{Cx¦}$ is shown to be equivalent to the existence of an n × n unitary matrix W such that $\text{A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{WB}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{+ B}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{W}{}^1\text{A}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{= 2C.}$ Some related inequalities are also discussed.     

Continuity of positive functionals on topological 1-algebras

It is shown that the set $\text{C}$_{m × n} of complex m × n matrices forms a lower semilattice under the partial ordering A ? B defined by $\text{A}{}^1\text{A = A}{}^1\text{B,}{}^1\text{AA}{}^1\text{= BA}{}^1\text{,}\text{where}\text{A}{}^1$ denotes the conjugate transpose of A. As a special case of a result for division rings, it is further shown that, over any field F, form = n = 2 and any proper involution 1 of F_{2 × 2}, the corresponding intersections A ∩ B all exist.     

Eigenvalue inequalities for products of matrix exponentials

Motivated by models from stochastic population biology and statistical mechanics, we proved new inequalities of the form $\text{(1) ?(e}{}^{\mathrm{A}}\text{e}{}^{\mathrm{B}}\text{)??(e}{}^{\mathrm{A+B}}\text{)}$, where A and B are n × n complex matrices, 1<n<∞, and ? is a real-valued continuous function of the eigenvalues of its matrix argument. For example, if A is essentially nonnegative, B is diagonal real, and ? is the spectral radius, then (1) holds; if in addition A is irreducible and B has at least two different diagonal elements, then the inequality (1) is strict. The proof uses Kingman's theorem on the log-convexity of the spectral radius, Lie's product formula, and perturbation theory. We conclude with conjectures.     

The asymptotic eigenvalue-distribution for a certain class of random matrices

For an n × n Hermitean matrix A with eigenvalues λ₁, …, λ_n the eigenvalue-distribution is defined by $\text{G(x, A) :=}\text{1}\text{n}$ · number {λ_i: λ_i ? x} for all real x. Let A_n for n = 1, 2, … be an n × n matrix, whose entries a_ik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, $\text{R}$, p) with the same distribution F_a. Suppose that all moments $\text{E}$ | a | ^k, k = 1, 2, … are finite, $\text{E}$a=0 and $\text{E}$ | a | ². Let

$\text{M(A)=}\text{∑}\text{σ=1}\text{s}\text{θ}{}_{\mathrm{\sigma }}\text{P}{}_{\mathrm{\sigma }}\text{(A,A}{}^1\text{)}$

with complex numbers θ_σ and finite products P_σ of factors A and $\text{A}{}^1$ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G₀(x) = G₁x) + G₂(x), where G₁ is a step function and G₂ is absolutely continuous, such that with probability $\text{1 G(x, M(}\text{A}{}_{\mathrm{n}}\text{n}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{))}$ converges to G₀(x) as n → ∞ for all continuity points x of G₀. The density g of G₂ vanishes outside a finite interval. There are only finitely many jumps of G₁. Both, G₁ and G₂, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples $\text{A}{}_{\mathrm{r}}\text{A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{+ A}{}^{\mathrm{1r}}$, $\text{A}{}^{\mathrm{r}}\text{A}{}^{\mathrm{1r}}\text{± A}{}^{\mathrm{1r}}\text{A}{}^{\mathrm{r}}$, r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form

$\text{lim sup}\text{n→∞}\text{∑}\text{i=1}{}^{\mathrm{n}}\text{|γ}{}_{\mathrm{i}}{}^{\mathrm{(n)}}\text{|}{}^2\text{?6A}{}_{\mathrm{n}}\text{6}{}^2\text{? 0.8228…}$

of Schur's inequality for the eigenvalues λ_i⁽ⁿ⁾ of A_n holds. Consequently random matrices do not tend to be normal matrices for large n.     

A note on polynomial matrix functions over a finite field

Let F_n denote the ring of n×n matrices over the finite field F=GF(q) and let A(x)=A_Nx^N+ ?+ A₁x+A₀?F_n[x]. A function $\text{?:F}{}_{\mathrm{n}}\text{→F}{}_{\mathrm{n}}$ is called a right polynomial function iff there exists an A(x)?F_n[x] such that $\text{?(B)=A}{}_{\mathrm{N}}\text{B}{}^{\mathrm{N}}\text{+?+A}{}_1\text{B+ A}{}_0$ for every B?F_n. This paper obtains unique representations for and determines the number of right polynomial functions.     

On the van der Waerden conjecture and zeros of polynomials

Let

$\text{F(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{A}{}_{\mathrm{n}}\text{≠ 0,}$

and

$\text{G(x) =}\text{∑}\text{k=o}\text{n}\text{n}\text{k}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}$

$\text{B}{}_{\mathrm{n}}\text{≠ 0,}$

be polynomials with real zeros satisfying A_n?1 = B_n?1 = 0, and let

$\text{H(x) =}\text{∑}\text{k=o}\text{n-2}\text{n}\text{k}\text{A}{}_{\mathrm{k}}\text{B}{}_{\mathrm{k}}\text{x}{}^{\mathrm{k}}\text{.}$

Using the recently proved validity of the van der Waerden conjecture on permanents, some results on the real zeros of H(x) are obtained. These results are related to classical results on composite polynomials.     

General scheme for solving linear algebraic problems by direct methods

Assume that A is an m × n real matrix with m?n and rank(A)=n. Let b be a real vector with m components and consider the problem of finding $\text{x=A}{}^2\text{b}$, where $\text{A}{}^2\text{=(A}{}^{\mathrm{T}}\text{A)}{}^{\mathrm{?1}}\text{A}{}^{\mathrm{T}}$. A general scheme for solving this problem is described. Many well-known and commonly used in practice direct methods can be found as special cases within the general scheme. This is illustrated by four examples. The general scheme can be used to study the common properties of the direct methods. The usefulness of this approach in the efforts to improve the efficiency of the direct methods for sparse matrices is demonstrated by formulating an algorithm that can be applied to any particular method belonging to the general scheme. This algorithm is implemented in several subroutines solving linear algebraic problems by different direct methods. Numerical results, obtained in a wide range of runs with these subroutines, are given.     

On the convergence of the Neumann series in interval analysis

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

On the matrix monotonicity of generalized inversion

Let A, B be two matrices of the same order. We write A>B(A>?B) iff A? B is a positive (semi-) definite hermitian matrix. In this paper the well-known result if

$\text{A>B>θ, then B}{}^{\mathrm{?1}}\text{>A}{}^{\mathrm{?1}}\text{> θ}$

(cf. Bellman [1, p.59]) is extended to the generalized inverses of certain types of pairs of singular matrices A,B?θ, where θ denotes the zero matrix of appropriate order.     

Concavity of certain maps on positive definite matrices and applications to Hadamard products

If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ₁ and Φ₂ are concave maps among positive definite matrices, then the following map involving tensor products:

$\text{(A,B)?f[Φ}{}_1\text{(A)}{}^{\mathrm{?1}}\text{?Φ}{}_2\text{(B)]·(Φ}{}_1\text{(A)?I)}$

is proved to be concave. If Φ₁ is affine, it is proved without use of positivity that the map

$\text{(A,B)?f[Φ}{}_1\text{(A)?Φ}{}_2\text{(B)}{}^{\mathrm{?1}}\text{]·(Φ}{}_1\text{(A)?I)}$

is convex. These yield the concavity of the map

$\text{(A,B)?A}{}^{\mathrm{1?p}}\text{?B}{}^{\mathrm{p}}$

(0<p?1) (Lieb's theorem) and the convexity of the map

$\text{(A,B)?A}{}^{\mathrm{1+p}}\text{?B}{}^{\mathrm{?p}}$

(0<p?1), as well as the convexity of the map

$\text{(A,B)?(A·log[A])?I?A?log[B]}$

.These concavity and convexity theorems are then applied to obtain unusual estimates, from above and below, for Hadamard products of positive definite matrices.     

Maximum zero strings of Bell numbers modulo primes

For any prime p, the sequence of Bell exponential numbers B_n is shown to have p ? 1 consecutive values congruent to zero (mod p), beginning with B_m, where $\text{m ≡ 1 ?}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}{}^2$ ($\text{mod}\text{(p}{}^{\mathrm{p}}\text{? 1)}\text{(p ? 1)}$). This is an improvement over previous results on the maximal strings of zero residues of the Bell numbers. Similar results are obtained for the sequence of generalized Bell numbers A_n generated by .     

Randomness implies order

Let τ: [0, 1] → [0, 1] possess a unique invariant density $\text{f}{}^1$. Then given any ? > 0, we can find a density function p such that $\text{∥ p ? f}{}^1\text{∥ < ?,}\text{and}\text{p}$ is the invariant density of the stochastic difference equation x_{n + 1} = τ(x_n) + W, where W is a random variable. It follows that for all starting points $\text{x}{}_0\text{? [0, 1],}\text{lim}{}_{\mathrm{n\to \infty }}\text{(1}\text{n)}\text{∑}{}_{\mathrm{i\; =\; 0}}{}^{\mathrm{n\; ?\; 1}}\text{χ}{}_{\mathrm{B}}\text{(x}{}_{\mathrm{i}}\text{) = ∝}{}_{\mathrm{B}}\text{p(ξ) dξ}$.     

Lower bounds for the norm of the symmetric product

Suppose each of m, n, and k is a positive integer, k ? n, A is a (real-valued) symmetric n-linear function on E_m, and B is a k-linear symmetric function on E_m. The tensor and symmetric products of A and B are denoted, respectively, by A ?B and A?B. The identity

$\text{6A · B6}{}^2\text{=}\text{∑}\text{q=0}\text{n}\text{(}\text{n}\text{k}\text{)}\text{(}\text{n+k}\text{k}\text{)}\text{6A?}{}_{\mathrm{q}}\text{B6}{}^2$

is proven by Neuberger in [1]. An immediate consequence of this identity is the inequality

$\text{6A · B 6}{}^2\text{?}\text{n+k}\text{n}{}^{\mathrm{?1}}\text{6A · B 6}{}^2$

In this paper a necessary and sufficient condition for

$\text{6A · B 6}{}^2\text{=}\text{n+k}\text{n}{}^{\mathrm{?}}\text{6A · B 6}{}^2$

is given. It is also shown that under certain conditions the inequality can be considerably improved. This improvement results from an analysis of the terms 6A?_qB6, 1?q?n, appearing in the identity.     

Barriers for uniformly elliptic equations and the exterior cone condition

We consider the mixed boundary value problem $\text{Au = f}\text{in}\text{Ω, B}{}_0\text{u = g}{}_0\text{in}\text{Γ}{}^{\mathrm{?}}\text{, B}{}_1\text{u = g}{}_1\text{in}\text{Γ}{}^{\mathrm{+}}$, where Ω is a bounded open subset of $\text{R}$ⁿ whose boundary Γ is divided into disjoint open subsets Γ⁺ and Γ^? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on $\text{Ω}$ and B_j, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on $\text{Γ}{}^{\mathrm{+}}$. The coefficients of the operators and Γ⁺, Γ^? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset $\text{T}$ of the reals such that, if $\text{D}{}^{\mathrm{s}}\text{= {u ? H}{}^{\mathrm{s}}\text{(Ω): Au = 0}}$ then for $\text{s ? =}\text{1}\text{2}\text{(}\text{mod}\text{1), (B}{}_0\text{,B}{}_1\text{): D}{}^{\mathrm{s}}\text{→ H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>1</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{?}}\text{) × H}{}^{\mathrm{<mtext>s\; ?\; </mtext><mtext>3</mtext><mtext>2</mtext>}}\text{(Γ}{}^{\mathrm{+}}\text{)}$ is a Fredholm operator if and only if s ∈$\text{T}$ . Moreover, $\text{T}$ = ?_xew$\text{T}$_x, where the sets $\text{T}$_x are determined algebraically by the coefficients of the operators at x. If n = 2, $\text{T}$_x is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, $\text{T}$_x is either an open interval of length 1 or is empty; and finally, if n ? 4, $\text{T}$_x is an open interval of length 1.     

Lower bounds for commutative radical Banach algebras

One says a commutative radical Banach algebra A has a lower bound if there is a lower growth condition on $\text{∥x}{}^{\mathrm{n}}\text{∥}{}^{\mathrm{<mtext>1</mtext><mtext>n</mtext>}}$ for all nonzero elements x in A. If A is a separable algebra we give necessary and sufficient conditions for A to possess a lower bound.     

Arnold's canonical matrices and the asymptotic simplification of ordinary differential equations

Let A(x,ε) be an n×n matrix function holomorphic for |x|?x₀, 0<ε?ε₀, and possessing, uniformly in x, an asymptotic expansion $\text{A(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{A}{}_{\mathrm{r}}\text{(x) ε}{}^{\mathrm{r}}$, as ε→0⁺. An invertible, holomorphic matrix function P(x,ε) with an asymptotic expansion $\text{P(x,ε)?Σ}{}^{\mathrm{\infty }}{}_{\mathrm{r=0}}\text{P}{}_{\mathrm{r}}\text{(x)ε}{}^{\mathrm{r}}$, as ε→0⁺, is constructed, such that the transformation y = P(x,ε)z takes the differential equation $\text{ε}{}^{\mathrm{h}}\text{dy}\text{dx}\text{= A(x,ε)y,h}$ a positive integer, into $\text{ε}{}^{\mathrm{h}}\text{dz}\text{dx}\text{= B(x,ε)z}$, where B(x,ε) is asymptotically equal, to all orders, to a matrix in a canonical form for holomorphic matrices due to V.I. Arnold.     

Interpolation and inner maps that preserve measure

It is shown, for n ? m ? 1, that there exist inner maps Φ: Bⁿ → B^m with boundary values $\text{Φ}{}_1\text{: B}{}^{\mathrm{n}}\text{→ B}{}^{\mathrm{m}}$ such that $\text{σ}{}_{\mathrm{m}}\text{(A) = σ}{}_{\mathrm{n}}\text{(Φ}{}_1{}^{\mathrm{?1}}\text{(A))}$. where σ_n and σ_m are the Haar measures on ?Bⁿ and ?B^m, respectively, and A ? Bⁿ is an arbitrary Borel set.