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Math416_SchurDecomposition - Math 416 Abstract Linear...

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Unformatted text preview: Math 416 - Abstract Linear Algebra Fall 2011, section E1 Schur decomposition Let us illustrate the algorithm to find a Schur decomposition, as in § 6.1, Theorem 1.1. Example: Find a Schur decomposition of the matrix A = 7- 2 12- 3 . Solution: First, we want an eigenvector of A . Let us find the eigenvalues: det( A- λI ) = 7- λ- 2 12- 3- λ = (7- λ )(- 3- λ ) + 24 = λ 2- 4 λ- 21 + 24 = λ 2- 4 λ + 3 = ( λ- 1)( λ- 3) . The eigenvalues are λ = 1 , 3. We could arbitrarily pick one of the two and find an eigenvector, but while we’re at it, let’s find both: λ = 1 : A- λI = A- I = 6- 2 12- 4 ∼ 3- 1 Take 1 3 , normalized to 1 √ 10 1 3 . λ = 3 : A- λI = A- 3 I = 4- 2 12- 6 ∼ 2- 1 Take 1 2 , normalized to 1 √ 5 1 2 . In fact, let’s pick λ 1 = 3 with normalized eigenvector u 1 = 1 √ 5 1 2 . We need to find an orthonormal basis { v 2 } of Span { u 1 } ⊥ = Span { 1 2 } ⊥ = Span {- 2 1 } . Pick v 2 = 1 √ 5- 2 1 ....
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Math416_SchurDecomposition - Math 416 Abstract Linear...

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