Schur Decomposition

The Schur decomposition of a complex square matrix is a matrix decomposition of the form

(1)

where is a unitary matrix, is its conjugate transpose, and is an upper triangular matrix which is the sum of a (i.e., a diagonal matrix consisting of eigenvalues of ) and a strictly upper triangular matrix .

Schur decomposition is implemented in the Wolfram Language for numeric matrices as SchurDecomposition[m]. The first step in a Schur decomposition is a Hessenberg decomposition. Schur decomposition on an matrix requires arithmetic operations.

For example, the Schur decomposition of the matrix

(2)

is

			(3)
			(4)

and the eigenvalues of are , , .