Definite

Let be an Inner Product Space (usually or )
A matrix is said to be definite if one of the following:

• positive definite:
• positive semi-definite:
• negative semi-definite:
• negative definite:

holds for all .
Note that is the element of the Dual Vector Space
corresponding to the Linear map .
When then .
When then .
We will write , , , and respectively
and often drop if its clear where the matrix lives.

Lemma

If is definite, then is Hermetian.

Lemma

If is symmetric and definite in , then is also definite in .