lecture_37

lecture_37 - MA 36600 LECTURE NOTES: FRIDAY, APRIL 23...

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Unformatted text preview: 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 6 = 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 = a 12 r- a 11 = ( r I- A ) = r- a 11- a 12- a 21 r- a 22 a 12 r- a 11 = p A ( r ) . In particular, when r = r 1 the right-hand side is the zero vector. This means we have eigenvectors (1) = a 12 r 1- a 11 and (2) = a 12 r 2- a 11 ....
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lecture_37 - MA 36600 LECTURE NOTES: FRIDAY, APRIL 23...

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