Unformatted text preview: r 1 and r 2 with corresponding eigenvectors ξ (1) = ° ξ 11 ξ 21 ± and ξ (2) = ° ξ 12 ξ 22 ± . Denote the 2 × 2 matrices D = ° r 1 r 2 ± and T = ° ξ 11 ξ 12 ξ 21 ξ 22 ± . The Frst matrix is constructed from the eigenvalues, whereas the second is constructed from the eigenvectors. We will show that T − 1 AT = D . To this end, recall the formula for the inverse of a matrix: ° ξ 11 ξ 12 ξ 21 ξ 22 ± − 1 = 1 ξ 11 ξ 22 − ξ 12 ξ 21 ° ξ 22 − ξ 12 − ξ 21 ξ 11 ± . Then we have the matrix product T − 1 AT = 1 ξ 11 ξ 22 − ξ 12 ξ 21 ° ξ 22 − ξ 12 − ξ 21 ξ 11 ± ° a 11 a 12 a 21 a 22 ± ° ξ 11 ξ 12 ξ 21 ξ 22 ± = 1 ξ 11 ξ 22 − ξ 12 ξ 21 ° ξ 22 − ξ 12 − ξ 21 ξ 11 ± ° r 1 ξ 11 r 2 ξ 12 r 1 ξ 21 r 2 ξ 22 ± = ° r 1 r 2 ± = D . 1...
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 Spring '09
 EdrayGoins
 Matrices, Howard Staunton, distinct eigenvalues, a12

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