William C. Waterhouse

The codimension of singular matrix pairs

Abstract The singular pairs of n × n matrices [those satisfying det( A − λB )  0] form a closed set of codimension n + 1 inside the space of all matrix pairs. The same holds for singular symmetric pairs. For Hermitian pairs, the singular ones form a closed set of codimension n + 1 or n + 2 according as n is odd or even. The irreducible components of these closed sets are determined by various basic singular summands.

One-Dimensional Affine Group Schemes*

The construction of these groups is straightforward; to an algebra B we assign the quotient R,,, G,/G,. The main effort comes in showing that these are the only possibilities. The key to this, and the basic technical idea in the paper, is the use of N&on blow-ups of group schemes over valuation rings. This process has been used before [ 1, 21 as a tool for resolving singularities, but in fact it also furnishes a good grasp on the structure of models in general. A further instance of this is found in the last section, where we give an analysis of all models (smooth or not) of G, over a valuation ring. It seems likely that the technique should also have other applications.

Do Symmetric Problems Have Symmetric Solutions

(1983). Do Symmetric Problems Have Symmetric Solutions? The American Mathematical Monthly: Vol. 90, No. 6, pp. 378-387.

Invertibility of linear maps preserving matrix invariants

There is a noninvertible linear map preserving a given function iff the function is constant on translates of a subspace. Any group preserving the function preserves the subspace. Hence it is often easy to isolate the cases in which such a noninvertible linear map can exist. For similarity invariants of matrices, this happens only for functions of the trace and for fuctions f satisfying f(X + λI) ≡ f(X).

The absolute-value estimate for symmetric multilinear forms☆

Abstract Let T ( v 1 , …, v n ) be a symmetric multilinear function on vectors v 1 , …, v n in some R r . Let M = max ‖ v ‖=1 | T ( v , …, v )|. Then | T ( v 1 , …, v n )| ⩽ ‖ v 1 ‖⋯‖ v n ‖ M . Equality (for nonzero v i ) can occur only in very special cases when the v i are not all parallel, and never if they span more than a plane. The inequality is known, but the proof here is new, as are the precise conditions for equality.

Gauss's first argument for least squares

Gauss first published an argument for the law of least squares in the Theoria Motus of 1809, though he had been using the law for some time.1 His argument there was influential throughout the nineteenth century, but historians recently have found its logic confusing. For instance, in the Sammelband prepared on the two-hundredth birthday of Gauss, Ivo Schneider describes the crucial point as follows:

Linear transformations preserving symmetric rank one matrices

Let 1’ be the space of n x n symmetric matrices over a commutative ring R (with unit), and let Y = Y(R) be the subset of rank < 1 (that is, the matrices where all 2 x 2 minors are zero). Our purpose is to study the invertible linear transformations z on V preserving Y. The study of such maps falls into two parts, the first of which has already been carried out: we know explicitly all the invertible T that formally preserve the vanishing of 2 x 2 minors. The basic maps of that kind are in fact just the ones that first come to mind, T(X) = ;1UXU” for invertible U and A. However, as I explained in [S], descent theory shows that these mappings (for even n) automatically produce certain others as well, and we call that larger set the natural transformations preserving rank d 1 (an explicit description of them is given in Section 1 below). More precisely, the theorem already known is this:

How often do determinants over finite fields vanish

Abstract The true formula is given for the probability that an n × n matrix over a finite field has determinant zero. This probability does not (as previously asserted) approach 1 as n grows.

Linear transformations on self-adjoint matrices: the preservation of rank-one-plus-scalar

Abstract Let M be the space of self-adjoint linear maps for a nondegenerate quadratic form on a finite-dimensional vector space (characteristic ≠ 2, dimension at least 3). Let T:M→M be an invertible linear transformation. If T preserves the set of rank-one-plus-scalar maps, then T is of the form T(X)=rPXP ∗ + ƒ(X)I (with one exception over Z ⧸3 Z ). Any T preserving commuting pairs also has this form (with no exceptions). The proof of this latter result involves computing the dimension of the space of self-adjoint maps commuting with any given one.

Similarity of matrices under SL(n,K)

Take a similarity class of n × n matrices over a field K Let pi ,(λ) m(i) be the elementary divisors Li , = K [λ]/(pi ). Under conjugation by SL(n, K), the class splits into subclasses corresponding to the elements of K×/Π(NL i ×) m(i).