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SchurDecomposition

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SchurDecomposition[m]
yields the Schur decomposition for a numerical matrix m, given as a list {q, t} where q is an orthonormal matrix and t is a block upper-triangular matrix.
SchurDecomposition[{m, a}]
gives the generalized Schur decomposition of m with respect to a.
  • The matrix m must be square.
  • For real-valued matrices m, setting the option RealBlockDiagonalForm->False allows complex values on the diagonal of the t matrix.
EXAMPLESCLOSE ALL
Basic Examples   (1)
Find the Schur decomposition of a real matrix:
Find the Schur decomposition of a real matrix:
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Scope   (3)
m is a 4×4 matrix with two real and two complex eigenvalues:
Find the Schur decomposition using machine arithmetic:
Find the Schur decomposition using 20-digit precision arithmetic:
Find the Schur decomposition of a matrix with complex entries:
m and a are 2×2 matrices:
Find the generalized Schur decomposition of m with respect to a:
Options   (2)
m is a 3×3 matrix:
With Pivoting->True, an extra matrix that represents the scaling and permutation is returned:
Verify that m.d is equal to d.q.t.Conjugate[Transpose[q]]:
m is a matrix with two real and two complex eigenvalues:
With RealBlockDiagonalForm->False, the result is complex upper triangular:
With RealBlockDiagonalForm->True, there are real 2×2 blocks along the diagonal:
Applications   (1)
m is a matrix that is not diagonalizable:
The Schur decomposition gives equivalence to a triangular matrix:
The eigenvalues, since they are real, occur on the diagonal of t:
Properties & Relations   (3)
m is a matrix with two real and two complex eigenvalues:
Find the Schur decomposition of m:
The real eigenvalues appear on the diagonal of t, the complex as a 2×2 block:
Verify m is equal to q.t.Conjugate[Transpose[q]]:
m and a are random 3×3 matrices:
Find the generalized Schur decomposition of m with respect to a:
Verify m is given by q.s.Conjugate[Transpose[p]]:
Verify a is given by q.t.Conjugate[Transpose[p]]:
m is a symmetric random matrix:
Find the Schur decomposition of m:
t is a diagonal matrix with the eigenvalues of m along the diagonal:
The columns of q are the eigenvectors of m:
Possible Issues   (1)
m is a matrix with exact integer valued entries:
SchurDecomposition only works with approximate numerical matrices:
For exact matrices, use JordanDecomposition:
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