"양의 정부호 행렬(positive definite matrix)"의 두 판 사이의 차이

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*  principal submatrix<br><math>\left( \begin{array}{c}  a_{1,1} \end{array} \right)</math>,<math>\left( \begin{array}{c}  a_{2,2} \end{array} \right)</math>,<math>\left( \begin{array}{c}  a_{3,3} \end{array} \right)</math><br><math>\left( \begin{array}{cc}  a_{1,1} & a_{1,2} \\  a_{2,1} & a_{2,2} \end{array} \right)</math>, <math>\left( \begin{array}{cc}  a_{1,1} & a_{1,3} \\  a_{3,1} & a_{3,3} \end{array} \right)</math>, <math>\left( \begin{array}{cc}  a_{2,2} & a_{2,3} \\  a_{3,2} & a_{3,3} \end{array} \right)</math><br><math>\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)</math><br>
 
*  principal submatrix<br><math>\left( \begin{array}{c}  a_{1,1} \end{array} \right)</math>,<math>\left( \begin{array}{c}  a_{2,2} \end{array} \right)</math>,<math>\left( \begin{array}{c}  a_{3,3} \end{array} \right)</math><br><math>\left( \begin{array}{cc}  a_{1,1} & a_{1,2} \\  a_{2,1} & a_{2,2} \end{array} \right)</math>, <math>\left( \begin{array}{cc}  a_{1,1} & a_{1,3} \\  a_{3,1} & a_{3,3} \end{array} \right)</math>, <math>\left( \begin{array}{cc}  a_{2,2} & a_{2,3} \\  a_{3,2} & a_{3,3} \end{array} \right)</math><br><math>\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)</math><br>
 
*  leading principal submatrix<br><math>\left( \begin{array}{c}  a_{1,1} \end{array} \right)</math><math>\left( \begin{array}{cc}  a_{1,1} & a_{1,2} \\  a_{2,1} & a_{2,2} \end{array} \right)</math>, <math>\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)</math><br>
 
*  leading principal submatrix<br><math>\left( \begin{array}{c}  a_{1,1} \end{array} \right)</math><math>\left( \begin{array}{cc}  a_{1,1} & a_{1,2} \\  a_{2,1} & a_{2,2} \end{array} \right)</math>, <math>\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)</math><br>
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<h5>예</h5>
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<math>\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)</math>
  
 
 
 
 

2011년 12월 14일 (수) 17:54 판

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개요
  • 실계수 n×n 행렬 M이 모든 0이 아닌 벡터 v 에 대하여, \( z^{T}M z > 0 \) 를 만족시킬 때, 양의 정부호 행렬이라 한다
  • 실베스터 판정법 - leading principal minor 가 모두 양수이면 양의 정부호 행렬이다

 

 

2×2 행렬의 경우
  • 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
    \(\left( \begin{array}{c} a_{1,1} \end{array} \right)\), \(\left( \begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{array} \right)\)

 

 

3×3 행렬의 경우
  • 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
    \(\left( \begin{array}{c} a_{1,1} \end{array} \right)\)\(\left( \begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{array} \right)\), \(\left( \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \end{array} \right)\)

 

 

 

예

\(\left( \begin{array}{ccccc} 2 & -1 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & -1 & 2 & -1 \\ 0 & 0 & 0 & -1 & 1 \end{array} \right)\)

 

 

역사

 

 

 

메모

 

 

 

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