rel="nofollow" href="#fb3_img_img_594d4282-fa25-5925-a273-e0a27cfc2419.png" alt="upper A"/>, denoted det
or
, is defined by
where
are referred to as the “cofactors” and are computed from
The term
is known as the “minor matrix” and is the matrix you get if you eliminate row
and column
from matrix
.
Finding the determinant depends on the dimension of the matrix
; determinants only exist for square matrices.
Example 2.6
For a 2 by 2 matrix
we have
Example 2.7
For a 3 by 3 matrix
we have
Definition 2.13 (Positive definite matrix) A square
matrix
is called positive definite if, for any vector
nonidentically zero, we have
Example 2.8
Let
be a 2 by 2 matrix
To show that
is positive definite, by definition
Therefore,
is positive definite.
Definition 2.14 (Positive semidefinite matrix) A matrix
is called positive semidefinite (or nonnegative definite) if, for any vector
, we have
Definition 2.15 (Negative definite matrix) A square
matrix
is called negative definite if, for any vector
nonidentically zero, we have
Example 2.9
Let
be a 2 by 2 matrix
To show that
is negative definite, by definition
