Jordan Matrix Decomposition

The Jordan matrix decomposition is the decomposition of a square matrix into the form

(1)

where and are similar matrices, is a matrix of Jordan canonical form, and is the matrix inverse of . In other words, is a similarity transformation of a matrix in Jordan canonical form. The proof that any square matrix can be brought into Jordan canonical form is rather complicated (Turnbull and Aitken 1932; Faddeeva 1958, p. 49; Halmos 1958, p. 112).

Jordan decomposition is also associated with the matrix equation and the special case .

The Jordan matrix decomposition is implemented in the Wolfram Language as JordanDecomposition[m], and returns a list s, j. Note that the Wolfram Language takes the Jordan block in the Jordan canonical form to have 1s along the superdiagonal instead of the subdiagonal. For example, a Jordan decomposition of

(2)

is given by

			(3)
			(4)