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


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: