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Date: 4-10-2021
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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.
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