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 $\text{A′ = A′}{}^1\text{such that}\text{A′B′ = B′A′}\text{and}\text{∥A ? A′∥? ?, ∥B ? B′∥? ?}$.     

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.     

On the fair coin tossing process

Let Ω = {1, 0} and for each integer n ≥ 1 let $\text{Ω}{}_{\mathrm{n}}\text{= Ω × Ω × … × Ω}$ (n-tuple) and $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}\text{= {(a}{}_1\text{, a}{}_2\text{, …, a}{}_{\mathrm{n}}\text{)|(a}{}_1\text{, a}{}_2\text{, … , a}{}_{\mathrm{n}}\text{) ? Ω}{}_{\mathrm{n}}\text{and}\text{Σ}{}_{\mathrm{i=1}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{i}}\text{= k}}$ for all k = 0,1,…,n. Let {Y_m}_m≥1 be a sequence of i.i.d. random variables such that $\text{P(Y}{}_1\text{= 0) = P(Y}{}_1\text{= 1) =}\text{1}\text{2}$. For each A in $\text{Ω}{}_{\mathrm{n}}$, let T_A be the first occurrence time of A with respect to the stochastic process {Y_m}_m≥1. R. Chen and A.Zame (1979, J. Multivariate Anal. 9, 150–157) prove that if n ≥ 3, then for each element A in $\text{Ω}{}_{\mathrm{n}}$, there is an element B in $\text{Ω}{}_{\mathrm{n}}$ such that the probability that T_B is less than T_A is greater than $\text{1}\text{2}$. This result is sharpened as follows: (I) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$, there is an element B also in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$ such that the probability that T_B is less than T_A is greater than $\text{1}\text{2}$; (II) for n ≥ 4 and 1 ≤ k ≤ n ? 1, each element A = (a₁, a₂,…,a_n) in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$, there is an element C also in $\text{Ω}{}_{\mathrm{n}}{}^{\mathrm{k}}$ such that the probability that T_A is less than T_C is greater than $\text{1}\text{2}$ if n ≠ 2m or n = 2m but a_i = a_{i + 1} for some 1 ≤ i ≤ n?1. These new results provide us with a better and deeper understanding of the fair coin tossing process.     

On majorization and Schur products

Suppose A, D₁,…,D_m are n × n matrices where A is self-adjoint, and let $\text{X = Σ}{}^{\mathrm{m}}{}_{\mathrm{k\; =\; 1}}\text{D}{}_{\mathrm{k}}\text{AD}{}^1{}_{\mathrm{k}}$. It is shown that if $\text{ΣD}{}_{\mathrm{k}}\text{D}{}^1{}_{\mathrm{k}}\text{= ΣD}{}^1{}_{\mathrm{k}}\text{D}{}_{\mathrm{k}}\text{= I}$, then the spectrum of X is majorized by the spectrum of A. In general, without assuming any condition on D₁,…,D_m, a result is obtained in terms of weak majorization. If each D_k is a diagonal matrix, then X is equal to the Schur (entrywise) product of A with a positive semidefinite matrix. Thus the results are applicable to spectra of Schur products of positive semidefinite matrices. If A, B are self-adjoint with B positive semidefinite and if b_ii = 1 for each i, it follows that the spectrum of the Schur product of A and B is majorized by that of A. A stronger version of a conjecture due to Marshall and Olkin is also proved.     

On the Schur multiplier norm of matrices

The Schur product of two n×n complex matrices A=(a_ij), B=(b_ij) is defined by A°B=(a_ijb_ij. By a result of Schur [2], the algebra of n×n matrices with Schur product and the usual addition is a commutative Banach algebra under the operator norm (the norm of the operator defined on $\text{C}$ⁿ by the matrix). For a fixed matrix A, the norm of the operator B?A°B on this Banach algebra is called the Schur multiplier norm of A, and is denoted by ∥A∥^m. It is proved here that $\text{∥A∥=∥U}{}^1\text{AU∥}{}_{\mathrm{m}}$ for all unitary U (where ∥·∥ denotes the operator norm) iff A is a scalar multiple of a unitary matrix; and that ∥A∥_m=∥A∥ iff there exist two permutations P, Q, a p×p (1?p?n) unitary U, an (n?p)×(n?p)1 contraction C, and a nonnegative number λ such that

$\text{A=λP}\text{U}\text{0}\text{0}\text{C}\text{Q;}$

and this is so iff $\text{∥A°}\text{A}\text{?}\text{∥=∥A∥}{}^2$, where ā is the matrix obtained by taking entrywise conjugates of A.     

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 .     

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.     

About some problems of spectral synthesis

For a given pair $\text{(A,b)∈}\text{R}{}^{\mathrm{n\times n}}\text{×}\text{R}{}^{\mathrm{n\times 1}}$ such that A is cyclic and b is a cyclic generator (with respect to A) of $\text{R}{}^{\mathrm{n\times 1}}$, it is shown that for every nonnegative integer m we can find a nonnegative integer t and a sequence $\text{{f}{}_{\mathrm{j\}}}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{,f}{}_{\mathrm{j}}\text{∈}\text{R}{}^{\mathrm{1\times n}}$,so that a the zeros of the rational function det P(z), where $\text{P(z) = zI ? A ? ∑}{}^{\mathrm{t}}{}_{\mathrm{j=0}}\text{z}{}^{\mathrm{-(m+j)}}\text{b?}$_f, lie in the open unit disc in the complex plane. The result is directly applicable to a stabilizability problem for linear systems with a time delay in control action.     

Nonnegative factorization of completely positive matrices

Let A be a real symmetric n × n matrix of rank k, and suppose that A = BB′ for some real n × m matrix B with nonnegative entries (for some m). (Such an A is called completely positive.) It is shown that such a B exists with $\text{m?}\text{1}\text{2}\text{k(k+1)?N}$, where 2N is the maximal number of (off-diagonal) entries which equal zero in a nonsingular principal submatrix of A. An example is given where the least m which works is $\text{(k+1)}{}^2\text{4}$ (k odd),$\text{k(k+2)}\text{4}$ (k even).     

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.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

More bounds for elgenvalues using traces

Let the n × n complex matrix A have complex eigenvalues λ₁,λ₂,…λ_n. Upper and lower bounds for Σ(Reλ_i)² are obtained, extending similar bounds for Σ|λ_i|² obtained by Eberlein (1965), Henrici (1962), and Kress, de Vries, and Wegmann (1974). These bounds involve the traces of A¹A, B², C², and D², where $\text{B}\text{=}\text{1}\text{2}\text{(}\text{A}\text{+}\text{A}{}^1\text{)}$, $\text{C}\text{=}\text{1}\text{2}\text{(}\text{A}\text{?}\text{A}{}^1\text{) /i}$, and $\text{D}\text{=}\text{AA}{}^1\text{?}\text{A}{}^1\text{A}$, and strengthen some of the results in our earlier paper “Bounds for eigenvalues using traces” in Linear Algebra and Appl. [12].     

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.     

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.     

A generalization of the fast LUP matrix decomposition algorithm and applications

We show that any m × n matrix A, over any field, can be written as a product, LSP, of three matrices, where L is a lower triangular matrix with l's on the main diagonal, S is an m × n matrix which reduces to an upper triangular matrix with nonzero diagonal elements when the zero rows are deleted, and P is an n × n permutation matrix. Moreover, L, S, and P can be found in O(m^α?1n) time, where the complexity of matrix multiplication is O(m^α). We use the LSP decomposition to construct fast algorithms for some important matrix problems. In particular, we develop O(m^α?1n) algorithms for the following problems, where A is any m × n matrix: (1) Determine if the system of equations $\text{A}\text{x}\text{=}\text{b}$ (where $\text{b}$ is a column vector) has a solution, and if so, find one such solution. (2) Find a generalized inverse, $\text{A}{}^1$, of A (i.e., $\text{AA}{}^1\text{A = A}$). (3) Find simultaneously a maximal independent set of rows and a maximal independent set of columns of A.     

Duality between some linear preserver problems. II. Isometries with respect to c-special norms and matrices with fixed singular values

Let $\text{F}{}_{\mathrm{m\times n}}$ (m?n) denote the linear space of all m × n complex or real matrices according as $\text{F}$=$\text{C}$ or $\text{R}$. Let c=(c₁,…,c_m)≠0 be such that c₁???c_m?0. The c-spectral norm of a matrix A?$\text{F}$_m×n is the quantity

$\text{6A6}{}_{\mathrm{c}}\text{∑}\text{i=I}\text{m}\text{c}{}_{\mathrm{i}}\text{σ}{}_{\mathrm{i}}\text{(A)}$

. where σ₁(A)???σ_m(A) are the singular values of A. Let d=(d₁,…,d_m)≠0, where d₁???d_m?0. We consider the linear isometries between the normed spaces $\text{(}\text{F}{}_{\mathrm{m\times }}{}_{\mathrm{n}}\text{,∥·∥}{}_{\mathrm{<mtext>c</mtext>}}\text{)}$ and $\text{(}\text{F}{}_{\mathrm{m\times }}{}_{\mathrm{n}}\text{,∥·∥}{}_{\mathrm{d}}\text{)}$, and prove that they are dual transformations of the linear operators which map $\text{L}$(d) onto $\text{L}$(c), where

$\text{L}\text{(c)= {X?}\text{F}{}_{\mathrm{m\times n}}\text{:X}\text{has singular values}\text{c}{}_1\text{,…,c}{}_{\mathrm{m}}\text{}}$

.     

The algebra of hyperboloids of revolution

If we change the sign of p ? m columns (or rows) of an m × m positive definite symmetric matrix A, the resultant matrix B has p negative eigenvalues. We give systems of inequalities for the eigenvalues of B and of the matrix obtained from B by deleting one row and column. To obtain these, we first develop characterizations of the eigenvalues of B which are analogous to the minimum-maximum properties of the eigenvalues of a symmetric A, i.e. the Courant-Fischer theorem. These results arose from studying probability distributions on the hyperboloid of revolution

$\text{x}{}^2{}_1\text{+ ? + x}{}^2{}_{\mathrm{m?p}}\text{? x}{}^2{}_{\mathrm{m\; ?\; p\; +\; 1}}\text{? ? ? x}{}^2{}_{\mathrm{m}}\text{= 1}$

. By contrast, the familiar results are associated with the sphere x²₁ + ? + x²_m = 1.     

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.     

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.