The generalized Eckart-Young principal axis theorem
David H. Nash · Linear and Multilinear Algebra · 1975
A rectangular matrix [apq ] is said to be diagonal if apq = 0 when p ≠ q. We present a simple proof of the following theorem of Wiegmann, but in principle given earlier by Eckart and Young: THEOREM If {Ai) is a set of complex r – s matrices such thatA A andA A are Hermitian for all i andj, then there exist unitary matrices P and Q such that for each i the matrixpA Q is real and diagonal Special cases of the above are well known and extremely useful. For example, the case n = 1 yields the classical singular value decomposition.