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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 I ) = 7  2 12 3 = (7 )( 3 ) + 24 = 2 4  21 + 24 = 2 4 + 3 = (  1)(  3) . The eigenvalues are = 1 , 3. We could arbitrarily pick one of the two and find an eigenvector, but while were at it, lets find both: = 1 : A I = A I = 6 2 12 4 3 1 Take 1 3 , normalized to 1 10 1 3 . = 3 : A I = A 3 I = 4 2 12 6 2 1 Take 1 2 , normalized to 1 5 1 2 . In fact, lets pick 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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This note was uploaded on 12/16/2011 for the course MATH 416 taught by Professor Frankland during the Fall '11 term at University of Illinois at Urbana–Champaign.
 Fall '11
 FRANKLAND
 Math, Linear Algebra, Algebra

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