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.