JordanReduce¶

Returns the Jordan canonical form of a square matrix (the Jordan matrix of JordanDecomposition, without the similarity matrix).

Distinct eigenvalues give a diagonal matrix in Eigenvalues order:

$ wo 'JordanReduce[{{1, 2}, {2, 1}}]'
{{3, 0}, {0, -1}}

Defective matrices show Jordan blocks with 1 on the superdiagonal; with a repeated eigenvalue the blocks sort ascending by eigenvalue:

$ wo 'JordanReduce[{{2, 1, 0}, {0, 2, 0}, {0, 0, 1}}]'
{{1, 0, 0}, {0, 2, 1}, {0, 0, 2}}

Complex and irrational eigenvalues stay exact:

$ wo 'JordanReduce[{{0, -1}, {1, 0}}]'
{{I, 0}, {0, -I}}

$ wo 'JordanReduce[{{1, 1}, {1, 0}}]'
{{(1 + Sqrt[5])/2, 0}, {0, (1 - Sqrt[5])/2}}