Math 2.0

Linear and multilinear algebra; matrix theory Special matrices Theorem

Eigenvalues of a positive definite matrix are positive

Builds on Positive definite matrixEigenvalue

Statement

Let AA be a positive definite n×nn \times n complex Hermitian matrix. Then each eigenvalue of AA is a positive real number.

Remarks

If Ax=λxAx = \lambda x with x≠0x \ne 0, then 0<x∗Ax=λ x∗x0 < x^* A x = \lambda\, x^* x, and x∗x>0x^* x > 0. The converse holds: a Hermitian matrix with only positive eigenvalues is positive definite, by the spectral theorem. So det⁡A>0\det A > 0 and tr⁡A>0\tr A > 0 for a positive definite AA.

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Matrix, 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 transposedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionPositive definite matrix, definition, no evidence levelPositive definite mat…definitionEigenvalues of a positive definite matrix are positive, statement, no evidence levelEigenvalues of a posi…statement
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  1. Matrix definition
  2. Vector space definition
  3. Square matrix definition
  4. Transpose definition
  5. Linear map definition
  6. Symmetric matrix definition
  7. Conjugate transpose definition
  8. Endomorphism definition
  9. Hermitian matrix definition
  10. Eigenvalue definition
  11. Positive definite matrix definition
  12. Eigenvalues of a positive definite matrix are positive statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Definite_matrix, Eigenvalues
About this page
Tag0HYC2RHD3M
URIm2://0HYC2RHD3M
Kindstatement, theorem
WritingW0 stub
MSC15B57, 15A18
StatusThis copy of the site has no build data. Run m2 status 0HYC2RHD3M.
UsesPositive definite matrix, Eigenvalue

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