Commuting pairs and triples of matrices and related varieties

In this note, we show that the set of all commuting d-tuples of commuting n×n matrices that are contained in an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also study the variety of n-dimensional commutative subalgebras of M_n(F), and show that it is irreducible of dimension n²−n for n4, but reducible, of dimension greater than n²−n for n7.     

Linear Operators that Preserve Commuting Pairs of Nonnegative Real Matrices

A pair of n×n matrices (A, B) is called a commuting pair if AB=BA. We characterize the linear operators that preserve the set of commuting pairs of matrices over a subsemiring of nonnegative real numbers.     

A characterization of linear transformations which leave the doubly stochastic matrices

A characterization of linear transformations which leave the n×n doubly stochastic matrices invariant is given as a linear combination of functions of the type T(X)=AXB with certain restrictions posed on the n×n matrices A and B.     

Chapter 3:inertia preservers

Let Vdenote either the space of n×n hermitian matrices or the space of n×nreal symmetric matrices, Given nonnegative integers r,s,t such that r+S+t=n, let G( r,s,r) denote the set of all matrices in V with inertia (r,s,t). We consider here linear operators on V which map G(r,s,t) into itself.     

The maximun of i(XAX-1+B)

Let F be an algebraically closed field. We denote by i(A) the number of invariant polynomials of a square matrix A, which are different from 1. For A,B any n×n matrices over F, we calculate the maximum of i(XAX^-1+B), where X runs over the set of all non-singular n×n matrices over F.     

Minimum permanents of doubly stochastic matrices with at least one zero entry

It is shown that the minimum value of the permanent on the n× ndoubly stochastic matrices which contain at least one zero entry is achieved at those matrices nearest to J_nin Euclidean norm, where J_nis the n× nmatrix each of whose entries is n^-1. In case n ≠ 3 the minimum permanent is achieved only at those matrices nearest J_n; for n= 3 it is achieved at other matrices containing one or more zero entries as well.     

Products of involutory matrices. I

It is shown that, for every integer ≥1 and every field F, each n×n matrix over F of determinant ±1 is the product of four involutory matrices over F. Products of three ×n involutory matrices over F are characterized for the special cases where n≤4 or F has prime order ≤5. It is also shown for every field F that every matrix over F of determinant ±1 having no more than two nontrivial invariant factors is a product of three involutory matrices over F.     

Finding all essential terms of a characteristic maxpolynomial

Let us denote ab=max(a,b) and ab=a+b for and extend this pair of operations to matrices and vectors in the same way as in linear algebra. We present an O(n²(m+n log n)) algorithm for finding all essential terms of the max-algebraic characteristic polynomial of an n×n matrix over with m finite elements. In the cases when all terms are essential, this algorithm also solves the following problem: Given an n×n matrix A and k{1,…,n}, find a k×k principal submatrix of A whose assignment problem value is maximum.     

Multiplicative properties of generalized matrix functions

We consider scalar-valued matrix functions for n×n matrices A=(a_ij) defined by Where G is a subgroup of S_n the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.

With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1).     

On the inertia of the Lyapunov transform AH+HA* for H >0

Given an arbitrary n×n matrix A with complex entries, we characterize all inertia triples (abc) that are attained by the Lyapunov transform AH+ HA^*, as H varies over the set of all n× n positive definite matrices.     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

Invariants of linear maps on matrix algebras

Let L be a linear map on the space of n×n matrices over a field. We determine the structure of the maps L that preserve commutativity. We also determine the structure of all linear maps on complex matrices that preserve the higher numerical range. The main tool is the Fundamental Theorem of Projective Geometry.     

Poincaré series of semi-invariants of 2×2 matrices

The Poincaré series of the algebra of -invariants of m-tuples of 2×2 matrices is presented both as a rational function and as a series of Schur functions. We show that this algebra of invariants is generated by the determinants, the mixed discriminants and the discriminants of 2×2 matrices. Consequences on invariants of three-dimensional matrices of the shape 2×2×m are discussed. For arbitrary n2, we prove an explicit functional equation for the Poincaré series of the -invariants of m-tuples of n×n matrices.     

Error analysis of QR decompositions by Givens transformations

The error analysis for computing the QR decomposition by Givens transformations was given originally by Wilkinson for n×n square matrices, and later by Gentleman for n×p (pn) tall thin matrices. The derivations were sufficiently messy that results were quoted by analogy to the derivation of a specific case. A certain lemma makes possible a much simpler derivation, which incidentally substantially tightens the bound. Moreover, it applies to variants of the method other than those originally considered, and suggests why observed errors are even less than this new bound.     

The structure of irreducible matrix groups with submultiplicative spectrum

We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p²× p² matrices if p is an odd prime.     

Caractérisation spectrale de la forme de Jordan

Soit , où désigne l'ensemble des matrices n×n à coefficients complexes. Nous montrons qu'on peut complètement caractériser la forme de Jordan de A en examinant le polynôme caractéristique de tA+X pour tous les tC et tous les . Ceci nous permet de donner une démonstration plus élémentaire d'un théorème de Baribeau et Ransford sur les transformations holomorphes de qui préservent le spectre.

Denote by the set of complex n×n matrices, and let . We give a variational, purely spectral characterization of the Jordan form of A by examining the characteristic polynomial of the perturbed matrices tA+X for tC and . This allows us to give a more elementary proof of a theorem of Baribeau and Ransford on spectrum-preserving holomorphic maps on .     

Manifolds of idemopotent matrices

Let M_n be the set of n×n matrices and r a nonnegative integer with r ≤ n. It is known,from Lie groups, that the rank r idempotent matrices in M_n form an arcwise connected 2n (n-r)-dimensional analytic manifold. This paper provides an elementary proof of this result making it accessible to a larger audience.     

The Best Lower Bound for the Permanent of a Correlation Matrix of Rank Two

If 1≤k≤n, then Cor(n,k) denotes the set of all n×n real correlation matrices of rank not exceeding k. Grone and Pierce have shown that if A∈Cor (n, n-1), then per(A)≥n/(n-1). We show that if A∈Cor(n,2), then , and that this inequality is the best possible.     

Minimum positive determinant of integer matrices with constant row and column sums

The least possible positive determinant of zero-one matrices that have constant row and column sums is determined, thus proving a conjecture of Newman. The result is extended to n×n integer matrices.     

Topological properties of orthostochastic matrices

The set of n×n orthostochastic matrices with the topology induced by the Euclidean matric is shown to be compact and path-connected. For n<3, the set of orthostochastic matrices is identical to the set of doubly stochastic matrices. In this paper, it is shown that for n3 the orthostochastic matrices are not everywhere dense in the set of doubly stochastic matrices, thus answering a question of L. Mirsky in his survey article on doubly stochastic matrices [2].