ExampleofOrthogonallyDiagonalize

ExampleofOrthogonallyDiagonalize - diagonalizable as it is...

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Example: Find a matrix P that orthogonally diagonalize A, and determine P - 1 AP , where A = 2 - 1 - 1 - 1 2 - 1 - 1 - 1 2 Solution: Find eigenvalues of A; Consider characteristic polynomial det ( λI - A ) = ± ± ± ± ± ± λ - 2 1 1 1 λ - 2 1 1 1 λ - 2 ± ± ± ± ± ± = λ ( λ - 3) 2 Hence λ = 0 and λ = 3 are eigenvalues of A. Find eigenspaces corresponding to λ = 0 and λ = 3 For this find nullspace of λI - A . For λ = 0 : Consider the homogenous system - Ax = 0 , from here we get the aug- mented matrix - 2 1 1 0 1 - 2 1 0 1 1 - 2 0 Row operations 1 - 2 1 0 0 3 - 3 0 0 0 0 0 From here we see that Solution space = { t (1 , 1 , 1) | t R } . Hence Basis for the eigenspace corresponding to λ = 0 is { (1 , 1 , 1) } For λ = 3 : Consider the homogenous system 3 I - Ax = 0 , from here we get the augmented matrix 1 1 1 0 1 1 1 0 1 1 1 0 Row operations 1 1 1 0 0 0 0 0 0 0 0 0 From here we see that Solution space = { t ( - 1 , 1 , 0) + s ( - 1 , 0 , 1) | t,s R } . Hence Basis for the eigenspace corresponding to λ = 3 is { ( - 1 , 1 , 0) , ( - 1 , 0 , 1) } . Note: Algebraic multiplicity and geometric multiplicity are the same for both eigen- values 0 and 3 respectively. This suggest that A is diagonalizable. Also A is orthogonally
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Unformatted text preview: diagonalizable as it is a symmetric matrix. Apply Gram-Schmidt process to the set { (1 , 1 , 1) , (-1 , 1 , 0) , (-1 , , 1) } We will form an orthonormal set: • v 1 = (1 , 1 , 1) • v 2 = (-1 , 1 , 0)-proj v 1 (-1 , 1 , 0) = (-1 , 1 , 0) • v 3 = (-1 , , 1)-proj v 1 (-1 , , 1)-proj v 2 (-1 , , 1) = (-1 2 ,-1 2 , 1) 1 • || v 1 || = √ v 1 · v 1 = √ 3 , || v 2 || = √ v 2 · v 2 = √ 2 , || v 3 || = √ v 3 · v 3 = √ 6 • { ( 1 √ 3 , 1 √ 3 , 1 √ 3 ) , (-1 √ 2 , 1 √ 2 , 0) , (-1 √ 6 ,-1 √ 6 , 2 √ 6 ) } is an orthonormal set. Form P : P = 1 √ 3-1 √ 2-1 √ 6 1 √ 3 1 √ 2-1 √ 6 1 √ 3 2 √ 6 Note that P-1 = P T hence it is an orthogonal matrix. Find D : P-1 AP = P T AP = D = 0 0 0 0 3 0 0 0 3 2...
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This document was uploaded on 04/26/2011.

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ExampleofOrthogonallyDiagonalize - diagonalizable as it is...

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