Math 2.0

Linear and multilinear algebra; matrix theory Definition

Singular value

Builds on Gram matrices are positive semidefiniteEigenvalue

Definition

Let AA be an m×nm \times n complex matrix. The matrix A∗AA^* A is positive semidefinite, so its eigenvalues λ1≥⋯≥λn≥0\lambda_1 \ge \cdots \ge \lambda_n \ge 0 are real and nonnegative. The singular values of AA are the numbers σi=λi(1≤i≤n).\sigma_i = \sqrt{\lambda_i} \qquad (1 \le i \le n).

Remarks

The number of nonzero singular values is the rank of AA. The largest singular value is the operator norm max⁡∥x∥=1∥Ax∥\max_{\norm{x} = 1} \norm{Ax}. For a normal matrix, the singular values are the absolute values of the eigenvalues. Some authors list only the first min⁡(m,n)\min(m, n) values.

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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 levelEigenvaluedefinitionPositive semidefinite matrix, definition, no evidence levelPositive semidefinite…definitionGram matrices are positive semidefinite, statement, no evidence levelGram matrices are pos…statementSingular value, definition, no evidence levelSingular valuedefinition
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  1. Matrix definition
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  4. Transpose definition
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  9. Eigenvalue definition
  10. Positive semidefinite matrix definition
  11. Gram matrices are positive semidefinite statement
  12. Singular value definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Singular_value, lead section
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DefinesSingular value
UsesGram matrices are positive semidefinite, Eigenvalue

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