Doubly stochastic matrices with prescribed positive spectrum

We show that for every set Λ={λ₁,λ₂,…,λ_n} of real numbers such that $\text{λ}{}_1\text{=1?λ}{}_2\text{???λ}{}_{\mathrm{n}}\text{\&\#62;0}$, there exists a doubly stochastic matrix with spectrum Λ. We present an explicit construction of such a matrix.     

Nonnegative matrices with prescribed spectrum and elementary divisors

In this paper we give new sufficient conditions for the existence and construction of nonnegative matrices with prescribed elementary divisors, which drastically improve and contain some of the previous known conditions. We also show how to perturb complex eigenvalues of a nonnegative matrix while keeping its nonnegativity. These results allow us, under certain conditions, to easily decide if a given list is realizable with prescribed elementary divisors.     

Decomposition of polynomial matrices into factors with prescribed canonical diagonal forms

We give a test for decomposability of a polynomial matrix over an arbitrary infinite field into factors with prescribed canonical diagonal forms whose product is the canonical diagonal form of the given matrix. We exhibit a method of actually constructing such decompositions of polynomial matrices.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 17–19.     

On orthogonal matrices with constant diagonal

In connection with the problem of finding the best projections of k-dimensional spaces embedded in n-dimensional spaces Hermann König asked: Given m∈$\text{R}$ and n∈$\text{N}$, are there n×n matrices C=(c_ij), i, j=1,…,n, such that c_ii=m for all i, |c_ij|=1 for i≠j, and C²=(m²+n?1)I_n? König was especially interested in symmetric C, and we find some families of matrices satisfying this condition. We also find some families of matrices satisfying the less restrictive condition CC^T=(m²+n?1)I_n.     

A note on “Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials”

Let F be any field and let B a matrix of F^q×p. Zaballa found necessary and sufficient conditions for the existence of a matrix A=[A_ij]_i,j∈{1,2}∈F^(p+q)×(p+q) with prescribed similarity class and such that A₂₁=B. In an earlier paper [A. Borobia, R. Canogar, Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials, Linear Algebra Appl. 424 (2007) 615-633] we obtained, for fields of characteristic different from 2, a finite step algorithm to construct A when it exists. In this short note we extend the algorithm to any field.     

On matrices stochastically similar to matrices with equal diagonal elements

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.     

On matrices stochastically similar to matrices with equal diagonal elements

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.     

Existence of matrices with prescribed entries

It is proved that, apart from for some exceptional cases, there always exists an n×n nonderogatory matrix over an arbitrary field with n prescribed entries and prescribed characteristic polynomial.     

Existence of matrices with prescribed eigenvalues and entries

It is proved improving a previous result of Oliveira that apart from two exceptions there always exists an n×n matrix with arbitarily prescribed 2n-3 entries and spectrum. Moreover, it is shown that the number 2n 3 of prescribed entries cannot be increased.     

Partial matrices all of whose completions have the same spectrum

We characterize the square partial matrices over a field all of whose completions have the same spectrum, and determine the maximum number of indeterminates in such partial matrices of a given order as well as the matrices that attain this maximum number.     

Products of matrices with prescribed spectra and ranks

This paper studies the possibility of writing a given square matrix as the product of two matrices with prescribed spectra and ranks. It extends some previously known results.     

Doubly stochastic matrices with some equal diagonal sums

Let A be doubly stochastic, and let τ₁,…,τ_m be m mutually disjoint zero diagonals in A, 1?m?n-1. E. T. H. Wang conjectured that if every diagonal in A disjoint from each τ_k (k=1,…,m) has a constant sum, then all entries in A off the m zero diagonals τ_k are equal to (n?m)^-1. Sinkhorn showed the conjecture to be correct. In this paper we generalize this result for arbitrary doubly stochastic zero patterns.     

Sums and products of matrices with prescribed similarity classes

Let F be a field and let A and n × n matrices over F. We study some properties of A^' + B^' and A^'B^', when A^' and B^' run over the sets of the matrices similar to A and B, respectively.     

Principal minors and diagonal similarity of matrices

Let A, B be n × n matrices with entries in a field F. Our purpose is to show the following theorem: Suppose n⩾4, A is irreducible, and for every partition of {1,2,…,n} into subsets α, β with ¦α¦⩾2, ¦β¦⩾2 either rank A[α¦β]⩾2 or rank A[β¦α]⩾2. If A and B have equal corresponding principal minors, of all orders, then B or B^t is diagonally similar to A.     

Constructing set systems with prescribed intersection sizes

Let f be an n-variable polynomial with positive integer coefficients, and let be a set system on the n-element universe. We define a set system and prove that f(H_i1∩H_i2∩∩H_ik)=|G_i1∩G_i2∩∩G_ik|, for any 1km, where f(H_i1∩H_i2∩∩H_ik) denotes the value of f on the characteristic vector of H_i1∩H_i2∩∩H_ik. The construction of is a straightforward polynomial-time algorithm from the set system and the polynomial f. In this paper we use this algorithm for constructing set systems with prescribed intersection sizes modulo an integer. As a by-product of our method, some upper bounds on the number of sets in set systems with prescribed intersection sizes are extended.     

Flows on graphs applied to diagonal similarity and diagonal equivalence for matrices

Three equivalence relations are considered on the set of n × n matrices with elements in F₀, an abelian group with absorbing zero adjoined. They are the relations of diagonal similarity, diagonal equivalence, and restricted diagonal equivalence. These relations are usually considered for matrices with elements in a field. But only multiplication is involved. Thus our formulation in terms of an abelian group with o is natural. Moreover, if F is chosen to be an additive group, diagonal similarity is characterized in terms of flows on the pattern graph of the matrices and diagonal equivalence in terms of flows on the bipartie graph of the matrices. For restricted diagonal equivalence a pseudo-diagonal of the graph must also be considered. When no pseudo-diagonal is present, the divisibility properties of the group F play a role. We show that the three relations are characterized by cyclic, polygonal, and pseudo-diagonal products for multiplicative F. Thus, our method of reducing propositions concerning the three equivalence relations to propositions concerning flows on graphs, provides a unified approach to problems previously considered independently, and yields some n, w or improved results. Our consideration of cycles rather than circuits eliminates certain restrictions (e.g., the complete reducibility of the matrices) which have previously been imposed. Our results extend theorems in Engel and Schneider [5], where however the group F is permitted to be non-commutative.     

Representation theorems and variational principles for self-adjoint operator matrices

We use the notion of triples D⁺ → H → D^? of Hilbert spaces to develop an analog of the Friedrichs extension procedure for a class of nonsemibounded operator matrices. In addition, we suggest a general approach (stated in the same terms) to the construction of variational principles for the eigenvalues of such matrices.