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A Primer on the Dirichlet Space by Thomas Ransford, Javad Mashreghi, Karim Kellay, Omar El-Fallah

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Appendix C

Positive definite matrices

C.1 Basic facts about positive definite matrices

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

image (C.1)

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 image 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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