an example for Schur decomposition
Let
A=(57-2-4). |
We will find an orthogonal matrix P and an upper triangular matrix
T such that PtAP=T applying the proof of Schur’s decomposition.
We ’re following the steps below
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We find the eigenvalues
of A
The eigenvalues of a matrix are precisely the solutions to the equationdet(λI-A)=0↔λ2-λ-6=0
Hence the roots of the quadratic equation (http://planetmath.org/QuadraticFormula) are the eigenvalues λ1=-2,λ2=3
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We find the eigenvectors
For each eigenvalue λi, solving the system(A-λiI)Xi=0 So we have that for λ1=-2
(A+2I)=0↔(77-2-2)(x1x2)=(00)→X1=(1,-1)
Analogously for λ2=3 the eigenvector X2=(7,-2)
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We get an orthonormal set
of eigenvectors using Gram-Schmidt orthogonalization
Consider the above two eigenvectors which are linearly independentbut are not orthogonal
X1 =(1,-1) X2 =(7,-2) First we take w1=X1=(1,-1). Therefore
w2=X2-w1⋅X2∥w1∥2w1 that is,
w2=(52,52) and finally the orthonormal set is {w1/∥w1∥,w2/∥w2∥}={(1√2,-1√2),(1√2,1√2)}
SoP=1√2(11-11). Then
T=PtAP=(-2903).
Title | an example for Schur decomposition![]() |
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Canonical name | AnExampleForSchurDecomposition |
Date of creation | 2013-03-22 15:27:02 |
Last modified on | 2013-03-22 15:27:02 |
Owner | georgiosl (7242) |
Last modified by | georgiosl (7242) |
Numerical id | 8 |
Author | georgiosl (7242) |
Entry type | Application |
Classification | msc 15-00 |
Related topic | SchurDecomposition |
Related topic | GramSchmidtOrthogonalization |