Math 2.0

Linear and multilinear algebra; matrix theory Definition

Rayleigh quotient

Builds on Hermitian matrixEigenvalue

Definition

Let AA be a Hermitian n×nn \times n complex matrix, and let x∈ℂnx \in \C^n with x≠0x \ne 0. The Rayleigh quotient of AA and xx is the real number R(A,x)=x∗Axx∗x.R(A, x) = \frac{x^* A x}{x^* x}.

Remarks

For each x≠0x \ne 0, λmin⁡≤R(A,x)≤λmax⁡\lambda_{\min} \le R(A, x) \le \lambda_{\max}, where λmin⁡\lambda_{\min} and λmax⁡\lambda_{\max} are the least and the greatest eigenvalues of AA. If xx is an eigenvector for λ\lambda, then R(A,x)=λR(A, x) = \lambda. So λmax⁡=max⁡x≠0R(A,x)\lambda_{\max} = \max_{x \ne 0} R(A, x) and λmin⁡=min⁡x≠0R(A,x)\lambda_{\min} = \min_{x \ne 0} R(A, x).

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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 mapdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionEigenvalue, definition, no evidence levelEigenvaluedefinitionRayleigh quotient, definition, no evidence levelRayleigh quotientdefinition
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  1. Matrix definition
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  4. Transpose definition
  5. Linear map definition
  6. Conjugate transpose definition
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  9. Eigenvalue definition
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Further reading

  1. Web https://en.wikipedia.org/wiki/Rayleigh_quotient, Bounds for Hermitian M
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MSC15A18, 15A42
DefinesRayleigh quotient
UsesHermitian matrix, Eigenvalue

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