On congruences for the traces of powers of some matrices

In a series of recent papers, V.I. Arnold studied many questions concerning the statistics and dynamics of powers of elements in algebraic systems. In particular, on the basis of experimental data, he proposed an Euler-type congruence for the traces of powers of integer matrices as a conjecture. The proof of this conjecture was deduced from the author’s theorem (obtained at the end of 2004) on congruences for the traces of powers of elements in number fields. Recently, it turned out that there also exist other approaches to congruences for the traces of powers of integer matrices. In the present paper, the author’s results of 2004 are strengthened and a survey of their relations to number theory, theory of dynamical systems, combinatorics, and p-adic analysis is given. The main conclusion of this survey is that all approaches considered here ultimately reflect different points of view on a certain simple but important phenomenon in mathematics.     

On the sum powers of matrices

A theorem of Lagrange says that every natural number is the sum of 4 squares. M. Newman proved that every integral n by n matrix is the sum of 8 (-1)ⁿ squares when n is at least 2. He asked to generalize this to the rings of integers of algebraic number fields. We show that an n by n matrix over a a commutative R with 1 is the sum of squares if and only if its trace reduced modulo 2Ris a square in the ring R/2R. It this is the case (and n is at least 2), then the matrix is the sum of 6 squares (5 squares would do when n is even). Moreover, we obtain a similar result for an arbitrary ring R with 1. Answering another question of M. Newman, we show that every integral n by n matrix is the sum of ten k-th powers for all sufficiently large n. (depending on k).     

On powers of nonnegative matrices

We investigate the structure of powers of nonnegative matrices A, and in particular characterize those A for which some power is (essentially) triangular. We also show how the number of irreducible components of A can be determined from its powers.     

Nonnegative matrices with stochastic powers

Matrices with nonnegative elements, which are nonstochastic but have stochastic powers, are considered. These matrices are characterized in the irreducible case and in the symmetric one. This paper represents part of a thesis submitted to the Senate of the Technion-Israel Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Science. The author wishes to thank Professor B. Schwarz for his guidance in the preparation of this paper.     

Spectra of matrices with P-matrix powers

We raise and partially answer the question of which sets of complex numbers can be the spectra of matrices all of whose powers are P-matrices. Several related questions are raised, and the partial results negatively resolve two earlier conjectures regarding spectra of P-matrices.     

On the convergence of powers of interval matrices

We give a necessary and sufficient condition for the sequence {$\text{A}$^k}of the powers of an interval matrix $\text{A}$ to converge to the null matrix $\text{O}$. We construct a point matrix $\text{B}$ which has spectral radius ? ($\text{B}$) less than one if {$\text{A}$^k}converges.     

Computing powers of arbitrary Hessenberg matrices

An efficient algorithm for the computation of powers of an n × n arbitrary lower Hessenberg matrix is presented. Numerical examples are used to show the computational details. A comparison of the algorithm with two other methods of matrix multiplication proposed by Brent and by Winograd is included. Related algorithms were proposed earlier by Datta and Datta for lower Hessenberg matrices with unit super-diagonal elements, and by Barnett for the companion matrix.     

On the invertlbility of hadamard powers of distance matrices

We determine the invertibilty or singularity, as appropriate, for all (positive) integral Hadamard powers of distance matrices.     

On the invertlbility of hadamard powers of distance matrices

We determine the invertibilty or singularity, as appropriate, for all (positive) integral Hadamard powers of distance matrices.     

On weak monotonicity of the powers of nonnegative matrices

The purpose of this paper is to study conditions under which the powers of a square matrix A are weak monotone. Necessary and sufficient conditions under which the product of any two weak monotone matrices is weak monotone are also obtained. Several examples are given to illustrate the necessity of the conditions stated.     

Doubly stochastic matrices whose powers eventually stop

In this note we characterize doubly stochastic matrices A whose powers A,A²,A³,… eventually stop, i.e., A^p=A^p+1= for some positive integer p. The characterization enables us to determine the set of all such matrices.     

Norms of powers of matrices with constrained spectra

If a matrix A of unit norm on n-dimensional Hilbert space has eigenvalues close to zero, then 6Aⁿ6 must be small. It is proved here that 6Aⁿ6 ?nr+O(r²), where r is the spectral radius of A, and the rate at which 6Aⁿ6 can approach 1 as r→1 is also ascertained [1?6Aⁿ6? O((1?r)^2n?1)]. More precise bounds for 6Aⁿ6 are obtained, some in terms of r and others in terms of all the eigenvalues of A.     

Positive powers of certain conditionally negative definite matrices

We investigate the spectrum of matrices (∣x_i,−x_j∣^a)ⁿ_i,j=1 with α>0 and distinct x₁,…,x_n whichare relevant to the theory of scattered data interpolation and spline functions. The main result is the non-singularity of these matrices, which is based on the property that the number of negativeand positive eigenvalues of these matrices is independent of x₁,…,x_n. Oscillation properties of asubset of eigenvectors of these matrices are also obtained. For 2<α<4 and points x¹,…,xⁿ∈R²,a sufficient condition for the non-singularity of (∥xⁱ−x^j∥^α₂)ⁿ_i,j=1 is derived.     

On fractional Hadamard powers of positive definite matrices

Let A = (a_ij) be a real symmetric n × n positive definite matrix with non-negative entries. We show that A^α ≡ (a_ij^α) is positive definite for all real α ? n ? 2. Moreover, the lower bound is sharp. We give related results for pairs of quadratic forms and discuss partial generalizations to the case in which A is a complex Hermitian matrix.     

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .     

Principal powers of matrices with positive definite real part

We study principal powers of complex square matrices with positive definite real part, or with numerical range contained in a sector. We extend the notion of geometric mean to such matrices and we establish an operator norm bound in this context.