Data Science in Theory and Practice. Maria Cristina Mariani

, 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

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