Today I Learned

Some of the things I've learned every day since Oct 10, 2016

57: Positive-Definite Matrices

A complex matrix A is said to be positive-definite iff for every column vector c \in \mathbb{C}^n,  c^{t} A c is real and positive. This is equivalent to the condition that all the eigenvalues of A are positive.

Similarly, there are variations of positive-definiteness with analogous conditions:

A positive-semidefinite \leftrightarrow c^{t} A c real and non-negative \leftrightarrow non-negative eigenvalues

A negative-semidefinite \leftrightarrow c^{t} A c real and non-positive \leftrightarrow non-positive eigenvalues

A negative-definite \leftrightarrow c^{t} A c real and negative \leftrightarrow negative eigenvalues

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