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 , , .

REFERENCES:

Golub, G. H. and Van Loan, C. F. Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, pp. 312-314, 1996.

Schur, I. "On the Characteristic Roots of a Linear Substitution with an Application to the Theory of Integral Equations." Math. Ann. 66, 488-510, 1909.