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Congruences and Norms of Hermitian Matrices

Two complex hermitian n × n matrices A and B are congruent if S*AS = B for some invertible n × n matrix S (with complex entries). The matrix S is called a congruence matrix. Given congruent hermitian matrices A and B, a congruence matrix is, of course, not unique. For instance, if A = B then one can take S = αI with |α| = 1, as well as any other matrix satisfying S*AS = A. However, here the choice S = I seems naturally to be best possible in the sense that when applied to an n-dimensional column vector it produces no distortion or movement of the vector at all. We shall measure the distortion (or movement) of the vector x ∊ C n under an n × n invertible matrix A in terms of ║x − Ax║, where the norm is euclidean. Then the distortion produced by A is ║I − A║, with the induced operator norm.