Schur Decomposition
المؤلف:
Golub, G. H. and Van Loan, C. F.
المصدر:
Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press
الجزء والصفحة:
...
29-9-2021
1499
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
![A=[3 2 1; 4 2 1; 4 4 0]](https://mathworld.wolfram.com/images/equations/SchurDecomposition/NumberedEquation2.gif) |
(2)
|
is
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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