David London

Eigenvalues of matrices with prescribed entries

It is shown that there exists an /i-square matrix all whose eigenvalues and n-1 of whose entries are arbitrarily prescribed. This result generalizes a theorem of L. Mirsky. It is also shown that there exists an «-square matrix with some of its entries prescribed and with simple eigenvalues, provided that n of the nonprescribed entries lie on a diagonal or, alternatively, provided that the number of prescribed entries does not exceed In—2. A well-known result of L. Mirsky [3] states essentially that, given any 2n—1 complex numbers Ax, ■ • • , Xn, ax, • • • , an_x, there exists an nsquare matrix with eigenvalues A,, • • • , Xn and n—\ of its main diagonal entries equal to ax, • • ■ , an_x. Related results for matrices over general fields were also obtained by Farahat and Ledermann [1]. We show that the restriction of the n—\ prescribed entries to the main diagonal is unnecessary. We first investigate the conditions under which there exists a matrix with prescribed entries and simple eigenvalues. A position in a matrix in which some entries have been prescribed is said to be free, if there is no prescribed entry in that position. By a diagonal in an «-square matrix we mean a set of n positions no two of which are in the same row or in the same column; i.e., positions (i, o(i)), /=1, •• • , n, for some permutation a. Theorem 1. Let ax, • • • , ani_n be n2—nprescribed complex numbers and let (it,ft), /= 1, ■ • • , n2—n, be prescribed different positions in an n-square matrix, such that the n remaining free positions form a diagonal of the matrix. Then there exists an n-square matrix with simple eigenvalues and with the prescribed entries at in the prescribed positions (it,jt), t= 1, • • • , n2—n. The number n of free positions cannot, in general, be decreased. Received by the editors June 1, 1971. AMS 1969 subject classifications. Primary 1525.

On matrices with a doubly stochastic pattern

In [7] Sinkhorn proved that if A is a positive square matrix, then there exist two diagonal matrices D, = {@r,..., d

Disconjugacy of complex differential systems and equations

) and D, = {dj2),..., di2r) with positive entries such that D,AD, is doubly stochastic. This problem was studied also by Marcus and Newman [3], Maxfield and Mint [4] and Menon [5]. Later Sinkhorn and Knopp [8] considered the same problem for A nonnegative. Using a limit process of alternately normalizing the rows and columns sums of A, they obtained a necessary and sufficient condition for the existence of D, and D, such that D,AD, is doubly stochastic. Brualdi, Parter and Schneider [l] obtained the same theorem by a quite different method using spectral properties of some nonlinear operators. In this note we give a new proof of the same theorem. We introduce an extremal problem, and from the existence of a solution to this problem we derive the existence of D, and D, . This method yields also a variational characterization for JJTC1 (din dj2’), which can be applied to obtain bounds for this quantity. We note that bounds for I7y-r (d,!‘) dj”)) may be of interest in connection with inequalities for the permanent of doubly stochastic matrices [3].

On the permanent of doubly stochastic matrices with zero diagonal

where W(z) = (wik(z))n. In [12], (1) was called disconjugate in D if, for every choice of n (not necessarily distinct) points z1, . . *, Znin D, the only solution of (1) which satisfies wi(zi) = 0, i = 1,..., n, is the trivial one w(z) 0_ . Disconjugacy of (1) in D is equivalent to the fact that for any fundamental solution W(z) = (wik(z))l of (2) (i.e., for any solution W(z) for which the determinant W(z) = wik(z)l#O for all z of D), the determinant Iwik(z?) In#0 for every choice of n (not necessarily distinct) points z1,..., Zn of D [12, Theorem 3]. (If this holds for one fundamental solution of (2), then it holds for all of them.) This property may thus serve as the definition of disconjugacy of the systems (2), and (1), in D. We define now a more restrictive property than this above defined (ordinary) disconjugacy in D. DEFINITION 1. The differential systems (1) and (2) are called zo-absolute disconjugate in D if there exists a point zo E D such that the solution W(z) = (wik(z)) of (2), determined by

On the Đoković conjecture for matrices of rank two

The permanent function on the set of n×n doubly stochastic matrices with zero main diagonaln⩽4, attains its minimum uniquely at the matrix whose off-diagonal entries are all equal to l/(n−1).

An inequality for the spectral norm of certain matrices

The Đokovic Conjecture asserts that here A is any doubly stochastic n × n matrix and pk (A) is the sum of the permanents of all k × k submatrices of A. We prove that if the case k = l of (∗) holds for all l × l, l  n, doubly stochastic matrices of rank two, then all the other cases of (∗) also hold for the same matrices. In a preceding paper a similar result was obtained for the generalized van der Waerden conjecture. In both cases the proof is based on a representation of p k(A) by means of a function (f(x)g(x)) of a pair of associated polynomials {f(x)g(x)}. Using this representation we also obtain some results on minimizing matrices corresponding to the van der Waerden and to the Dokovic conjectures. Finally we outline some properties of the function (f(x)g(x))

On the Van Der Waerden conjecture for matrices of rank two

Let , be nonnegativc numbers such that , where . Let U = (uij ) and V = (uij ) matrices, let denote the spectral matrix norm. We prove that if , then Inequality (∗) was conjectured by B. Schwarz and A. Zaks.

Nonnegative matrices with stochastic powers

The generalized van der Waerden conjecture asserts that Here Ais any doubly stochastic n×n matrixJn is the n×n matrix all whose entries are equal to 1/n and pk (A) is the sum of the permanents of all k×k submatrices of A. The casek=n of (*) is the (ordinary) van der Waerden conjecture.It is proved that if the van der Waerden conjecture holds for all k×k k⩽n doubly stochastic matrices of rank two, then the generalized conjecture also holds for the same matrices. The proof is based on a representation of pk (A) by means of an associated pair of polynomials.

On matrices stochastically similar to matrices with equal diagonal elements

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.

Diagonals of matrices stochastically similar to a given matrix

A necessary and sufficient condition for a matrix to be stochastically similar to a matrix with equal diagonal elements is obtained Aand B are called Stochastically similar if B=SAS − 1 where S is quasi-stochastic i.e., all row sums of .S are I. An inverse elementary divisor problem for quasi-stochastic matrices is also considered.