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  • Symmetric matrices

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  • Definition of symmetric matrix
  • Symmetric matrices are square.
  • Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal.
  • Definition of orthogonally diagonalizable matrix
  • A matrix is orthogonally diagonalizable if and only if it is symmetric.
  • The spectral theorem for symmetric matrices
  • Formula for the spectral decomposition for a symmetric matrix
  • Definition of skew-symmetric matrix
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  1. The spectral theorem for symmetric matrices

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  • A matrix is orthogonally diagonalizable if and only if it is symmetric.Close match
  • Formula for the spectral decomposition for a symmetric matrixClose match
  • Symmetric matricesClose match
  • Eigenvectors of a symmetric matrix with different eigenvalues are orthogonal.Close match
  • Definition of orthogonally diagonalizable matrixClose match
  • A complex matrix is unitarily diagonalizable iff it is normalClose match
  • Normal matricesClose match
  • A complex normal matrix has an orthonormal eigenvector basisClose match
  • Matrix diagonalizationClose match
  • Hermitian matricesClose match
  • A square real matrix has orthonormal columns iff its inverse is its transposeClose match
  • Definition of skew-symmetric matrixClose match
  • Definition of orthogonal matrixClose match
  • Every complex square matrix is unitarily similar to an upper triangular matrixClose match
  • Unitary matrices preserve orthogonal (orthonormal) bases.Close match
  • Hermitian matrices have real eigenvalues.Close match
  • Definition of symmetric matrixClose match
  • Distinct eigenvalues of a Hermitian matrix have orthogonal eigenvectors.Close match
  • An n-by-n matrix is diagonalizable if and only if the union of the basis vectors for the eigenspaces is a basis for R^n (or C^n).Close match
  • Symmetric matrices are square.Close match

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