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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 deFne 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 . ±irst 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 (1) = 11 21 and (2) = 12 22 . Denote the 2 2 matrices D = r 1 r 2 and T = 11 12 21 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: 11 12 21 22 1 = 1 11 22 12 21 22 12 21 11 . Then we have the matrix product T 1 AT = 1 11 22 12 21 22 12 21 11 a 11 a 12 a 21 a 22 11 12 21 22 = 1 11 22 12 21 22 12 21 11 r 1 11 r 2 12 r 1 21 r 2 22 = r 1 r 2 = D . 1...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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