Math 2.0

Linear and multilinear algebra; matrix theory Bilinear and quadratic forms Theorem

Sylvester's criterion

Builds on Positive definite matrixMinor

Statement

Let AA be an n×nn \times n complex Hermitian matrix. For 1≤k≤n1 \le k \le n, let Δk\Delta_k be the determinant of the upper left k×kk \times k submatrix of AA. Then AA is positive definite if and only if Δk>0\Delta_k > 0 for all k=1,…,nk = 1, \dots, n.

Remarks

The numbers Δk\Delta_k are the leading principal minors of AA. For (abbc)\begin{pmatrix} a & b \\ b & c \end{pmatrix} real, the test is a>0a > 0 and ac−b2>0ac - b^2 > 0. For positive semidefinite matrices, the analogous test needs all principal minors, not only the leading ones, to be nonnegative.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionSymmetric matrix, definition, no evidence levelSymmetric matrixdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionPositive definite matrix, definition, no evidence levelPositive definite mat…definitionDeterminant, definition, no evidence levelDeterminantdefinitionMinor, definition, no evidence levelMinordefinitionSylvester's criterion, statement, no evidence levelSylvester's criterionstatement
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Transpose definition
  6. Linear map definition
  7. Symmetric matrix definition
  8. Conjugate transpose definition
  9. Multilinear map definition
  10. Hermitian matrix definition
  11. Alternating multilinear map definition
  12. Positive definite matrix definition
  13. Determinant definition
  14. Minor definition
  15. Sylvester's criterion statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Sylvester's_criterion, lead section
About this page
Tag0WKEF78PMV
URIm2://0WKEF78PMV
Kindstatement, theorem
WritingW1 stated
MSC15B57, 15A15
StatusThis copy of the site has no build data. Run m2 status 0WKEF78PMV.
UsesPositive definite matrix, Minor
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