Math 2.0

Linear and multilinear algebra; matrix theory Canonical forms and decompositions Theorem

Polar decomposition

Builds on Unitary matrixSquare root of a positive semidefinite matrix

Statement

Let AA be an n×nn \times n complex matrix. Then there are a unitary matrix UU and a positive semidefinite Hermitian matrix PP with A=UP.A = UP. The factor PP is unique: P=(A∗A)1/2P = (A^* A)^{1/2}, the square root of A∗AA^* A. If AA is invertible, then UU is also unique.

Remarks

It follows from the singular value decomposition A=WΣV∗A = W \Sigma V^*: take U=WV∗U = W V^* and P=VΣV∗P = V \Sigma V^*. For n=1n = 1, it is the polar form z=eiθrz = e^{i\theta} r of a complex number. There is also a left polar decomposition A=P′UA = P' U with P′=(AA∗)1/2P' = (A A^*)^{1/2}.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionPositive semidefinite matrix, definition, no evidence levelPositive semidefinite…definitionSquare root of a positive semidefinite matrix, definition, no evidence levelSquare root of a posi…definitionPolar decomposition, statement, no evidence levelPolar decompositionstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Identity matrix object
  2. Matrix definition
  3. Square matrix definition
  4. Transpose definition
  5. Conjugate transpose definition
  6. Hermitian matrix definition
  7. Unitary matrix definition
  8. Positive semidefinite matrix definition
  9. Square root of a positive semidefinite matrix definition
  10. Polar decomposition statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Polar_decomposition, lead section
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Tag0J2C7BDJQA
URIm2://0J2C7BDJQA
Kindstatement, theorem
WritingW1 stated
MSC15A23
StatusThis copy of the site has no build data. Run m2 status 0J2C7BDJQA.
UsesUnitary matrix, Square root of a positive semidefinite matrix
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