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Schur Decomposition


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

 Q^(H)AQ=T=D+N,
(1)

where Q is a unitary matrix, Q^(H) is its conjugate transpose, and T is an upper triangular matrix which is the sum of a D=diag(lambda_1,lambda_2,...,lambda_n) (i.e., a diagonal matrix consisting of eigenvalues lambda_i of A) and a strictly upper triangular matrix N.

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 n×n matrix requires O(n^3) arithmetic operations.

For example, the Schur decomposition of the matrix

 A=[3 2 1; 4 2 1; 4 4 0]
(2)

is

Q=[0.49857 0.76469 0.40825; 0.57405 0.061628 -0.81650; 0.64953 -0.64144 0.40825]
(3)
T=[6.6056 4.4907 -0.82632; 0.00000 -0.60555 1.0726; 0.00000 0.00000 -1.00000],
(4)

and the eigenvalues of T are lambda_1=-1, lambda_2=3-sqrt(13)=-0.60555..., lambda_3=3+sqrt(13)=6.6055....


See also

Eigenvalue, Hessenberg Decomposition, Matrix Decomposition

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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. "Über die charakteristischen Wurzeln einer linearen Substitution mit einer Anwendung auf die Theorie der Integralgleichungen." Math. Ann. 66, 488-510, 1909.

Referenced on Wolfram|Alpha

Schur Decomposition

Cite this as:

Weisstein, Eric W. "Schur Decomposition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SchurDecomposition.html

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