Math 2.0

Linear and multilinear algebra; matrix theory Inner product spaces Theorem

Spectral theorem for normal matrices

Builds on Normal matrixUnitary matrix

Statement

Let AA be an n×nn \times n complex matrix. Then AA is normal if and only if there are a unitary matrix UU and a diagonal matrix DD with A=UDU∗.A = U D U^*.

Remarks

The columns of UU are then an orthonormal basis of ℂn\C^n of eigenvectors of AA. If AA is Hermitian, then DD is real; if AA is unitary, then the diagonal entries of DD have absolute value 11. Over ℝ\R, a normal matrix need not be diagonalisable: a rotation of the plane is normal.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionNormal matrix, definition, no evidence levelNormal matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionSpectral theorem for normal matrices, statement, no evidence levelSpectral theorem for …statement
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. Transpose definition
  4. Conjugate transpose definition
  5. Normal matrix definition
  6. Unitary matrix definition
  7. Spectral theorem for normal matrices statement

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_matrix, Consequences
About this page
Tag04CW1P44VD
URIm2://04CW1P44VD
Kindstatement, theorem
WritingW0 stub
MSC15A21, 47B15
StatusThis copy of the site has no build data. Run m2 status 04CW1P44VD.
UsesNormal matrix, Unitary matrix

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