The simultaneous diagonalization of two real symmetric (r.s.) matrices has long been of interest. This subject is generalized here to the following problem (this question was raised by Dr. Olga Taussky-Todd, my thesis advisor at the California Institute of Technology) : What is the first simultaneous block diagonal structure of a nonsingular pair of r.s. matrices ? For example, given a nonsingular pair of r.s. matrices S and T, which simultaneous block diagonalizations X'SX = diag(A 1 ..... Ak), X'TX = diag(B v ..., Bk) with dim A i = dim B i and X nonsingular are possible for 1 ~ k ~ n; and how well defined is a simultaneous block diagonalization for which k, the number of blocks, is maximal ? Here a pair o f r.s. matrices S and T is called nonsingular if S is nonsingular.
If the number of blocks k is maximal, then one can speak of the finest simultaneous block diagonalization of S and T, since then the sizes of the blocks A i are uniquely determined (up to permutations) by any set of generators of the pencil P(S, T) = {aS + bTla, b e ~} via the real Jordan normal form of S-IT. The proof uses the canonical pair form theorem for nonsingular pairs of r.s. matrices. The maximal number k and the block sizes dim A i are also determined by the factorization over