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MA 36600 LECTURE NOTES: FRIDAY, APRIL 23 Diagonalizing 2 × 2 Matrices Distinct Eigenvalues. We explain how we chose the matrices D and T in the previous example. Indeed, we discuss a general theory which holds especially well for 2 × 2 matrices. Consider such a matrix: A = a 11 a 12 a 21 a 22 . Recall that we can define the trace , determinant , and discriminant : tr A = a 11 + a 22 , det A = a 11 a 22 a 12 a 21 ; disc A = (tr A ) 2 4 (det A ) = ( a 11 a 22 ) 2 + 4 a 12 a 21 . We assume that disc A = 0. Then p A ( r ) = r 2 (tr A ) r + (det A ) has two distinct roots r 1 and r 2 . First we give formulas for the eigenvectors of such a matrix when the eigenvalues are known. Observe that if we choose ξ = a 12 r a 11 = ( r I A ) ξ = r a 11 a 12 a 21 r a 22 a 12 r a 11 = 0 p A ( r ) . In particular, when r = r 1 the right-hand side is the zero vector. This means we have eigenvectors ξ (1) = a 12 r 1 a 11 and ξ (2) = a 12 r 2 a 11 . This gives a rather simple formula for eigenvectors of a 2 × 2 matrix A when two distinct eigenvalues r 1 and r 2 are known. Next we give formulas for the matrices D and T . More generally, say that we have distinct eigenvalues
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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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