Positive definite Hermitian matrices and reproducing kernels

Abstract

Some quadratic identities associated with positive definite Hermitian matrices are derived by use of the theory of reproducing kernels. For example, the following identity is obtained: Let{ A j } m j=1 be N × N positive definite Hermitian matrices. Then, for any complex vector x ∈ C N , we have the identity x ∗ ∑ j= 1 m A -1 j -1 x = min ∑ j= 1 m x ∗ jA j x j . The minimum is taken here over all the decompositions x =∑ m j=1 x j . This identity gives, in a sense, a precise converse for an inequality which was derived by T. Ando. Moreover, this paper shows that the sum of two reproducing kernels is naturally related to the harmonic-arithmetic-mean inequality for matrices and also that the geometric-arithmetic-mean inequality for matrices can be naturally interpreted in terms of tensor-product spaces.