Definiteness Examples
Leading minors Example
This is an example of evaluating the leading minor of a matrix, as we explained on the previous page. Given the following matrix $A$,
\[ A = \begin{pmatrix} 4 & 2 & 2 \\ 2 & 10 & 7 \\ 2 & 7 & 21 \\ \end{pmatrix} \]
The leading minors of A are:
- $det(\Delta_1) = 4$
- $det(\Delta_2) = 4 * 10 - 2 * 2 = 36$
- $det(\Delta_3)$
- $= 4 * (10 * 21 - 7 * 7) - 2 * (2* 21 -2 * 7) + 2 * ( 2 * 7 - 2 * 10)$
- $= 4 * 161 - 68 = 3 * 161 + 93$
- $= (3*16)*10 + 100 - 7 + 3 = 576$
Every leading minor is greater than $0$, so the matrix is positive definite.