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# Q52 - Math 330 Quiz 5.2 Determine the eigenvalues and...

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Unformatted text preview: Math 330 Quiz 5.2 Determine the eigenvalues and eigenspaces of the matrix A: 3 0 0 A= 0 1 2 0 2 1 Is there a basis for R2 consisting of eigenvectors for A? SOLUTION: |A - I| = 3- 0 0 0 1- 2 0 2 1- = (3 - ) 1- 2 2 1- = (3 - ) (1 - )2 - 4 = (3 - ) 2 - 2 + 1 - 4 = (3 - )(2 - 2 - 3) = (3 - )( - 3)( + 1) So the eigenvalues are = 3 and = -1. To find the eigenspaces we proceed with each eigenvalue separately: = -1 We examine the null space of A + I: 4 0 0 (R1,R2)( R1 , R2 ) 1 0 0 4 0 0 R3R3-R2 4 2 0 2 2 0 1 1 = = A+I = 0 2 2 0 0 0 0 0 0 0 2 2 x1 So with an associated eigenvector of x = x2 we have that x3 x1 0 x1 = 0 x2 = x3 -1 x2 = -x3 x3 1 0 So the 1-eigenspace is the span of -1 . 1 =3 Here we look at the null space of A - 3I: 0 0 0 0 0 0 R2 R2 0 0 0 R3R3+R2 -2 0 -2 2 = 0 1 -1 A - 3I = 0 -2 2 = 0 2 -2 0 0 0 0 0 0 x1 So our eigenvector x = x2 must satisfy just the single equation x2 = x3 . Thus both x1 and x3 are free x3 variables and hence the 3-eigenspace is the span of 0 1 0 , 1 0 1 So we do indeed have a basis for R3 consisting of eigenvectors for A. 1 ...
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