Find the Schur decomposition of a real matrix:

Find the Schur decomposition of a machine-precision matrix:

Format the results:

Schur decomposition of a complex matrix:

Consider a real-valued matrix with complex eigenvalues:

Compute its Schur decomposition:

All entries in the decomposition are real:

The matrix is block-upper triangular, with entries along the first subdiagonal, but not upper triangular:

RealBlockDiagonalFormFalse makes upper triangular at the cost of complex entries:

Schur decomposition of an arbitrary-precision matrix:

Schur decomposition of sparse matrices:

Schur decomposition a structured matrix of type SymmetrizedArray:

Generalized Schur decomposition for a pair of matrices m and a:

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:

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 that m is equal to q.t.ConjugateTranspose[q]:

m and a are random 3×3 matrices:

Find the generalized Schur decomposition of m with respect to a:

Verify that m is given by q.s.ConjugateTranspose[p]:

Verify that a is given by q.t.ConjugateTranspose[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:

m is a matrix with exact integer valued entries:

SchurDecomposition only works with approximate numerical matrices:

For exact matrices, use JordanDecomposition: