Olga Taussky

Pairs of matrices with property

This note is concerned, for matrices with elements in an algebraically closed field of arbitrary characteristic p, with pencils generated by pairs of matrices with property L. A pair of n by n matrices is said to have property L if for a special ordering of the characteristic roots a< of A and Bi of B, the characteristic roots of \A+ptB are Xa<+p,/3,- for all values of X and pt. (See [1-5].) In §§1-5 another characterization of pairs of matrices with property L is given for a large class of such pairs. The method employed for this purpose is used in §6 for the study of pencils (not necessarily with property L) of diagonable matrices, i.e., matrices which are similar to a diagonal matrix. (These matrices are also called nondefective.) It is shown that for p=0, as well as for n^p, such pencils are always generated by commutative matrices. In §7 the significance of this result for general pencils of commutative matrices is investigated. 1. The ^-discriminant. The new characterization of pairs A, B of matrices with property L is obtained by considering those ratios X/p. for which \A +p,B has a multiple characteristic root. We see as follows that this is the case either for at most n(n — l) ratios or for every X/p.

L

which has the roots 0, -3, 5. The root 0 is contained in all three circles. It can however be shown that an analogue of the situation for n = 2 holds if further conditions are imposed on the elements aik. Theorem 1. The dominant root of a matrix of positive elements canno t be a common point of all n circles Oi unless it is a common boundary point of at least two of the circles. Proof. It is known [2] that the dominant root A of such a matrix is real and positive and that the corresponding characteristic vector XI, ... , Xn can be chosen in such a way that all its components are posi tive. Consider then the equation

. II

Do you wonder how Miss Noether herself would react to this celebration in her honor? We can imagine her turning or tilting her head to one side, considering the import of it all and smiling shyly, but proudly, her eyes bright behind those thick lenses. She would be listening intently to these algebra talks, never missing a word, perhaps a little breathless with concentrated interest.

Bounds for characteristic roots of matrices

Proof. Let ~ denote the space of all ant isymmetric n x n matrices with elements in F. The space ~ has dimension n(n--1)/2, and, since char F=~2, we have 6 ~ = { 0 } . Hence 6 and ~ together span the whole linear space ~. Choose a basis B x . . . . . 13,, of ~ such that B x . . . . . B~r are in 6 , and BN+ 1 . . . . . B, , are in ~. With respect to this basis (ordered in a row) the matr ix of g has the form

Emmy Noether in Bryn Mawr

A certain triple diagonal matrix was studied extensively by Mark Kac in connection with problems in statistical mechanics. It had been considered earlier by Sylvester and Schrodinger and later by Siegert and Hess. This paper, which is of an expository and historical character, contains some new matrix intensive proofs of the results of Kac and Rozsa, as well as proofs of related binomial coefficient identities of Taussky.

On the matrix function Ax+X′A′

The triangular factorization of symmetric matrices, usually ascribed to A. L. Cholesky, was (essentially) explicitly given by O. Toeplitz earlier, but rather parenthetically. Actually, the word ‘matrix’ does not appear in the report on Cholesky’s process; the matrix formulation seems due to Henry Jensen.

Another look at a matrix of Mark Kac

Some time ago in our public library I picked up a book, The Best of All Possible Worlds, by Peter F. Drucker, a professor of social science at Claremont Graduate School. In that book, Drucker writes about a pupil of Riemann who was to write his thesis on quaternions. Riemann had seen their importance to his own areas of study, and his student saw that they would lead to the subject that we now call matrix algebra, which has become all the rage. But matrices were not always the rage. They have played a large role in group theory since the work of Elie Cartan, and they play a role in physics and in statistics. Still, matrix theory reached me only slowly. Since my main subject was number theory, I did not look for matrix theory. It somehow looked for me. In what follows a number of instances of such events are sketched.

Cholesky, Toeplitz and the triangular factorization of symmetric matrices

Abstract The expression of det ( AB-BA ) as a norm in both (Q( m ) and Q( n ) for 2×2 rational matrices with characteristic roots in Q( m ) , resp. Q( n ) , is studied here further, see [1]. A necessary and sufficient condition for this element to be also a norm in Q( m , n ) is obtained.

How I became a torchbearer for matrix theory

There is a natural link between classes of ideals in orders of algebraic number fields and similarity classes of integral matrices defined by unimodular matrices. Two fractional ideals in an order of an algebraic number field are called arithmetically equivalent if and only if they differ by a factor in the field. It is known that the number of classes obtained in this way is finite and that the classes form a finite abelian semigroup. In order to study and generalize these ideal classes orders in finite extensions of more general fields are considered. In order to describe the results obtained several abstract concepts concerning semigroups are introduced: An element a of a multiplicative semigroup S is called invertible if the equations

Additive commutators of rational 2×2 matrices☆

In this paper ideal matrices with respect to ideals in the maximal order of an algebraic number field are connected with the different of the field and with group matrices in the case of normal fields whose maximal order has a normal basis.