Math 2.0

statement theorem

Spectral theorem for Hermitian matrices

Statement

Let AA be a Hermitian n×nn \times n complex matrix. Then there are a unitary matrix UU and a real diagonal matrix DD with A=UDU∗.A = U D U^*. If AA is a real symmetric matrix, then UU can be chosen real, that is, orthogonal: A=QDQTA = Q D Q^T.

Remarks

The diagonal entries of DD are the eigenvalues of AA, and the columns of UU are orthonormal eigenvectors. This is the matrix form of the spectral theorem. For example, (2112)=Q(3001)QT\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix} = Q \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix} Q^T with Q=12(111−1)Q = \frac{1}{\sqrt 2} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}.

Explanations

L1 human

Every Hermitian matrix is diagonal after a unitary change of coordinates. The new coordinate axes are perpendicular eigenvectors. For real symmetric matrices, this is a rotation, possibly with a reflection, of the axes.

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Identity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionBasis, definition, no evidence levelBasisdefinitionEigenvector, definition, no evidence levelEigenvectordefinitionOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionSpectral theorem, statement, no evidence levelSpectral theoremstatementSpectral theorem for Hermitian matrices, statement, no evidence levelSpectral theorem for …statement
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  1. Identity matrix object
  2. Matrix definition
  3. Vector space definition
  4. Square matrix definition
  5. Transpose definition
  6. Inner product definition
  7. Linear combination definition
  8. Linear map definition
  9. Linear subspace definition
  10. Conjugate transpose definition
  11. Inner product space definition
  12. Orthogonal vectors definition
  13. Linear independence definition
  14. Endomorphism definition
  15. Linear span definition
  16. Hermitian matrix definition
  17. Unitary matrix definition
  18. Orthonormal family definition
  19. Adjoint of a linear map definition
  20. Eigenvalue definition
  21. Spanning set definition
  22. Self-adjoint operator definition
  23. Basis definition
  24. Eigenvector definition
  25. Orthonormal basis definition
  26. Spectral theorem statement
  27. Spectral theorem for Hermitian matrices statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Spectral_theorem, Hermitian maps and Hermitian matrices

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