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  • Eigensystem[ m ] gives a list values , vectors of the eigenvalues and eigenvectors of the square matrix m.
  • Eigensystem finds numerical eigenvalues and eigenvectors if m contains approximate real numbers.
  • The elements of m can be complex.
  • All the non-zero eigenvectors given are independent. If the number of eigenvectors is equal to the number of non-zero eigenvalues, then corresponding eigenvalues and eigenvectors are given in corresponding positions in their respective lists.
  • If there are more eigenvalues than independent eigenvectors, then each extra eigenvalue is paired with a vector of zeros.
  • Eigensystem[ m , ZeroTest -> test ] applies test to determine whether expressions should be assumed to be zero. The default setting is ZeroTest -> Automatic.
  • The eigenvalues and eigenvectors satisfy the matrix equation m.Transpose[ vectors ] == Transpose[ vectors ].DiagonalMatrix[ values ].
  • See the Mathematica book: Section 3.7.9.
  • See also: NullSpace, JordanDecomposition, SchurDecomposition, QRDecomposition.
  • Related packages: LinearAlgebra`Orthogonalization`, LinearAlgebra`Cholesky`.

    Further Examples

    These are the eigenvalues of a 2 x 2 matrix.



    These are the eigenvectors of the matrix.



    This lists the eigenvalues and the eigenvectors for the matrix. The eigenvalues are listed first.



    This verifies the result for the first eigenvalue and its corresponding eigenvector.



    Using Eigensystem to diagonalize a matrix
    Matrices with non-repeated eigenvalues can be diagonalized using Eigensystem. The eigenvalues give the diagonalized form. The columns of the change of basis matrix b are the eigenvectors.



    We check the diagonalization.