양의 정부호 행렬(positive definite matrix)

이 항목의 수학노트 원문주소

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