**Definition C.1.1** A complex *n* × *n* matrix *A* = (*a*_{jk}) is called *positive semidefinite* if, for any choice of complex numbers *λ*_{1}, . . . , *λ*_{n}, we have

(C.1) |

It is *positive definite* if, further, equality holds only when all the *λ*_{j} 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, ...

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