Definition C.1.1 A complex n × n matrix A = (ajk) is called positive semidefinite if, for any choice of complex numbers λ1, . . . , λn, we have
It is positive definite if, further, equality holds only when all the λj are zero.
It is easy to check that, if condition (C.1) holds, then for all j, k Thus, if A is positive semi-definite, then it is automatically a hermitian matrix.
Now let A be an n × n hermitian matrix. According to the spectral theorem, ...