Positive-definite Matrix - Block Matrices

Block Matrices

A positive 2n × 2n matrix may also be defined by blocks:

where each block is n × n. By applying the positivity condition, it immediately follows that A and D are hermitian, and C = B*.

We have that z*Mz ≥ 0 for all complex z, and in particular for z = ( v, 0)T. Then

A similar argument can be applied to D, and thus we conclude that both A and D must be positive definite matrices, as well.