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