On the maximal number of linearly independent real vectors annihilated simultaneously by two real quadratic forms
Frank Uhlig · Pacific Journal of Mathematics · 1973
For a nonsingular pair of real symmetric (r.s.) matrices S and T the maximal number m of lin.ind.vectors simultaneously annihilated by the associated quadratic forms is computed as a function of the real Jordan normal form of S~*T.Conversely one can deduce which real Jordan normal form S" 1 T must have, if a specific m is the maximal number of such vectors.Furthermore, two new conditions are found that assure S and T to be simultaneously diagonalizable by a real congruence transformation.First we introduce the notions of Jordan blocks, real Jordan normal form and the canonical pair form for pairs of r.s.matrices.DEFINITION 0.1.A square matrix of the form ιX e 0 M = \θ X k x k is called a Jordan block of type (A), if for k ^ 2 we have λ e R and e -1, while for k -1 we have M -(λ) with XeR.Such a matrix M is called a Jordan block of type (B), if for k ^ 4 we have X -(& ~" a)> α ' b e R > b φ ° and e = (o l)' while for k = 2 we have M ^ ~~ ) with a, beR, b Φ 0. Jordan blocks will also be denoted by J(λ, k) and J(a, b, k) y respectively.Now we can state the real Jordan normal form theorem (see, e.g., Kowalski [2], p. 248).THEOREM 0.1.Every real square matrix A is similar over the reals to a matrix J ~ diag (A u, A t )> in which each square block A ά corresponds to an eigenvalue Xj of A. If this eigenvalue Xj is real, the associated A 3 -is a Jordan block of type (A); if X 3 -= a + bi 0 R, then Aj is a Jordan block of type (B).This J is called the real Jordan normal form of A. It is uniquely determined by A, except for the order of its Jordan blocks.