Math 2.0

statement theorem

Eigenvalues of a positive definite matrix are positive

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.

Explanations

L1 human

A positive definite matrix stretches each of its eigenvector directions by a positive factor. It never flips or crushes a direction. So its eigenvalues, its determinant and its trace are all positive.

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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

Sources

  1. Web https://en.wikipedia.org/wiki/Definite_matrix, Eigenvalues

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