A Study of the Canonical Form for a Pair of Real Symmetric Matrices and Applications to Pencils and to Pairs of Quadratic Forms

A pair of real symmetric matrices S and T is called a nonsingular pair if Sis nonsingular. A new treatment for obtaining the classical canonical pair form for a nonsingular pair is obtained by the use of results on commuting matrices and by elementary matrix algebra. This canonical form is used to obtain formulas for an arbitrary real n X n matrix A that relate the dimensions of both the space N of real I symmetric matrices T such that AT = TA and the space of products AT such that AT is symmetric to the real Jordan normal form of A. The first formula expresses a previously found result in a simpler way while the second one is new. These formulas are then applied to prove anew the known result that A is nonderogatory iff dim N = n. Simultaneous diagonalization of two real symmetric matrices has been of interest. For instance it has been shown that if the quadratic forms associated with Sand T (of dimensions greater than 2) do not vanish simultaneously, then S and T can be diagonalized simultaneously by a real congruence transformation. This subject is generalized here to the study of the following two problems: 1) The finest simultaneous block diagonal structure for nonsingular pairs, 2) common annihilating vectors of the corresponding quadratic forms. The proofs are obtained here by algebraic means. Results: ad 1) A simultaneous block diagonalization X' TX= diag(A1,...,Ak and X'TX = diag(B1,...,Bk) with dim Ai = dim Bi and X nonsingular is the finest simultaneous block diagonalization of a nonsingular pair S and T, if k is maximal. In this finest diagonalization the sizes of the blocks Ai are uniquely determined (up to permutations) by any set of generators of the pencil P(S,T) = {aS + bT]a,b ϵ R}. The number k and the sizes of the diagonal blocks are also derived from the factorization over C of f(λ,µ) = det(λS + µT) for λ, µ ϵ R. ad 2) Knowing the real Jordan normal form of S-1T for a nonsingular pair S and T we compute the maximal number m of linearly independent vectors that are simultaneously annihilated by the corresponding quadratic forms. Conversely, knowing m for two quadratic forms we deduce the first simultaneous block diagonal structure of S and T, the corresponding pair of real symmetric matrices. This is used to give new sufficient conditions for S and T to be simultaneously diagonalizable.