`SchurDecomposition`¶

Gives the real Schur decomposition {q, t} of a square machine-precision matrix: q is orthogonal, t is quasi-upper-triangular (a 2×2 block for each complex pair of eigenvalues), and m == q . t . Transpose[q].

A matrix that is already quasi-upper-triangular is its own Schur form:

$ wo 'SchurDecomposition[N[{{1, 2}, {0, 3}}]]'
{{{1., 0.}, {0., 1.}}, {{1., 2.}, {0., 3.}}}

$ wo 'SchurDecomposition[N[{{0, 1}, {-1, 0}}]]'
{{{1., 0.}, {0., 1.}}, {{0., 1.}, {-1., 0.}}}

The factors rebuild the matrix, and the diagonal of t carries the eigenvalues:

$ wo 'a = N[{{4, 1, 0}, {1, 3, 1}, {0, 1, 2}}]; {q, t} = SchurDecomposition[a]; Chop[q . t . Transpose[q] - a]'
{{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

$ wo 'Round[Sort[Diagonal[Last[SchurDecomposition[N[{{4, 1, 0}, {1, 3, 1}, {0, 1, 2}}]]]]], 0.0001]'
{1.2679, 3., 4.7321}

An exact matrix has no machine-precision Schur form:

$ wo 'SchurDecomposition[{{1, 2}, {3, 4}}]'

SchurDecomposition::schurf: SchurDecomposition has received a matrix with infinite precision.
SchurDecomposition[{{1, 2}, {3, 4}}]

SchurDecomposition¶

`SchurDecomposition`¶