양의 정부호 행렬(positive definite matrix)

행렬

$\left( \begin{array}{cc} a_{1,1} & a_{1,2} \ a_{2,1} & a_{2,2} \end{array} \right)$
principal submatrix

$\left( \begin{array}{c} a_{1,1} \end{array} \right)$ , $\left( \begin{array}{c} a_{2,2} \end{array} \right)$ , $\left( \begin{array}{cc} a_{1,1} & a_{1,2} \ a_{2,1} & a_{2,2} \end{array} \right)$
leading principal submatrix

$\left( \begin{array}{c} a_{1,1} \end{array} \right)$ , $\left( \begin{array}{cc} a_{1,1} & a_{1,2} \ a_{2,1} & a_{2,2} \end{array} \right)$

행렬

$\left( \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \ a_{2,1} & a_{2,2} & a_{2,3} \ a_{3,1} & a_{3,2} & a_{3,3} \end{array} \right)$
principal submatrix

$\left( \begin{array}{c} a_{1,1} \end{array} \right)$ , $\left( \begin{array}{c} a_{2,2} \end{array} \right)$ , $\left( \begin{array}{c} a_{3,3} \end{array} \right)$

$\left( \begin{array}{cc} a_{1,1} & a_{1,2} \ a_{2,1} & a_{2,2} \end{array} \right)$ , $\left( \begin{array}{cc} a_{1,1} & a_{1,3} \ a_{3,1} & a_{3,3} \end{array} \right)$ , $\left( \begin{array}{cc} a_{2,2} & a_{2,3} \ a_{3,2} & a_{3,3} \end{array} \right)$

$\left( \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \ a_{2,1} & a_{2,2} & a_{2,3} \ a_{3,1} & a_{3,2} & a_{3,3} \end{array} \right)$
leading principal submatrix

$\left( \begin{array}{c} a_{1,1} \end{array} \right)$ $\left( \begin{array}{cc} a_{1,1} & a_{1,2} \ a_{2,1} & a_{2,2} \end{array} \right)$ , $\left( \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \ a_{2,1} & a_{2,2} & a_{2,3} \ a_{3,1} & a_{3,2} & a_{3,3} \end{array} \right)$

다음과 같은 5x5 행렬을 생각하자

$\left( \begin{array}{ccccc} 2 & -1 & 0 & 0 & 0 \ -1 & 2 & -1 & 0 & 0 \ 0 & -1 & 2 & -1 & 0 \ 0 & 0 & -1 & 2 & -1 \ 0 & 0 & 0 & -1 & 1 \end{array} \right)$
leading principal submatrix와 그 행렬식을 구하면 다음과 같다

$\begin{array}{ll} \left( \begin{array}{c} 2 \end{array} \right) & 2 \ \left( \begin{array}{cc} 2 & -1 \ -1 & 2 \end{array} \right) & 3 \ \left( \begin{array}{ccc} 2 & -1 & 0 \ -1 & 2 & -1 \ 0 & -1 & 2 \end{array} \right) & 4 \ \left( \begin{array}{cccc} 2 & -1 & 0 & 0 \ -1 & 2 & -1 & 0 \ 0 & -1 & 2 & -1 \ 0 & 0 & -1 & 2 \end{array} \right) & 5 \ \left( \begin{array}{ccccc} 2 & -1 & 0 & 0 & 0 \ -1 & 2 & -1 & 0 & 0 \ 0 & -1 & 2 & -1 & 0 \ 0 & 0 & -1 & 2 & -1 \ 0 & 0 & 0 & -1 & 1 \end{array} \right) & 1 \end{array}$