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.     

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.     

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.     

Lie products in incidence algebras

We give conditions when a strictly upper triangular element of an incidence algebra over a commutative ring is the Lie commutator of two elements of the incidence algebra, one of which is strictly upper triangular. In particular, it follows that this is the case for the ring of n × n upper triangular matrices, where n is either finite or infinite.     

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.     

Self-duality results for unitary and hermitian matrices

It is known that, if T is an n × n complex matrix such that every characteristic root of UT has modulus I for every n × n unitary matrix U then T must be unitary. This paper generalizes this result in two directions, one of which provides a proof of a 1971 conjecture of M. Marcus. An analogous self-duaiity result is given for hermitian matrices, and several additional results of self-duality type are given concerning hermitian matrices and real matrices, using the trace and the determinant.     

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.     

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).     

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.     

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.     

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.     

A note on commuting pairs of matrices

We give a short proof of the Motzkin-Taussky result that the variety of commuting pairs of matrices is irreducible. An easy consequence of this is that any two generated commutative subalgebra of n×n matrices has dimension at most n. We also answer an old question of Gerstenhaber by showing that the set of commuting triples of n×n matrices is not irreducible for n≥32.     

On the extreme eigenvalues of hermitian (block) toeplitz matrices

We are concerned with the behavior of the minimum (maximum) eigenvalue λ₀⁽ⁿ⁾ (λ_n⁽ⁿ⁾) of an (n + 1) × (n + 1) Hermitian Toeplitz matrix T_n(ƒ) where ƒ is an integrable real-valued function. Kac, Murdoch, and Szegö, Widom, Parter, and R. H. Chan obtained that λ₀⁽ⁿ⁾ — min ƒ = O(1/n^2k) in the case where ƒ C^2k, at least locally, and ƒ — inf ƒ has a zero of order 2k. We obtain the same result under the second hypothesis alone. Moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices, proving that the rate of convergence of λ₀⁽ⁿ⁾ to inf ƒ depends only on the order ρ (not necessarily even or integer or finite) of the zero of ƒ — inf ƒ. With the help of this tool, we derive an absolute lower bound for the minimal eigenvalues of Toeplitz matrices generated by nonnegative L¹ functions and also an upper bound for the associated Euclidean condition numbers. Finally, these results are extended to the case of Hermitian block Toeplitz matrices with Toeplitz blocks generated by a bivariate integrable function ƒ.     

On the maximum of per(I-A)

The purpose of this paper is to give some necessary conditions on a substochastic matrix which maximizes the values of Per(I-A) for A taken in the semigroup of n × n substochastic matrices, and to determine the exact value of the maximum which is found to be 2^[n/2].     

Nonconvexity of the permanent

Counterexamples are given which show that for n≥4 the permanent is not convex on the n by n correlation matrices.     

Minimum Permanents on a Face of the Polytope of Doubly Stochastic Matrices II

We determine the minimum permanents and minimizing matrices on the face of z_3+n, for the fully indecomposable (0,1) matrices of order 3+n, which include an identity submatrix of order n .     

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.     

Additive maps preserving M-P inverses of matrices over Fields

Suppose F is a field of characteristic not 2 or 3. A characterization is given for all additive maps, on the algebra of all n × n matrices over F. which preserve Moore -Penrose(M-P) Inverses of 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].