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lecture_38 (dragged) - MA 36600 LECTURE NOTES MONDAY APRIL...

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Unformatted text preview: MA 36600 LECTURE NOTES: MONDAY, APRIL 27 Repeated Eigenvalues Jordan Canonical Form. We explain the general theory for when we can diagonalize a 2 × 2 matrix A. Say that we have eigenvalues r1 and r2 . i. If r1 ￿= r2 i.e., we have distinct eigenvalues, then there exist 2 × 2 matrices ￿ ￿ ￿ ￿ ξ ξ r 0 T = 11 12 and D= 1 such that T−1 A T = D. ξ21 ξ22 0 r2 ii. If r1 = r2 i.e., we have repeated eigenvalues, then there exist 2 × 2 matrices ￿ ￿ ￿ ￿ ξ11 η1 r1 c T= and J= such that T−1 A T = J. ξ21 η2 0 r1 Here, either c = 0 or c = 1. In fact, c = 0 if and only if A = r1 I is a diagonal matrix. Note that A is diagonalizable if and only if either (1) A has distinct eigenvalues or (2) A = r I is a scalar multiple of the identity matrix. The matrices D and J are called the Jordan canonical form for A. We sketch the proof of this result. Denote the eigenvectors of r1 and r2 as ￿￿ ￿￿ ξ ξ ξ (1) = 11 and ξ (2) = 12 . ξ21 ξ22 We have three cases to consider: Case #1: r1 ￿= r2 . We saw in the previous lecture that ￿ ￿ ￿ ￿ ξ ξ rξ r2 ξ12 T = 11 12 =⇒ A T = 1 11 = TD ξ21 ξ22 r1 ξ21 r2 ξ22 Case #2: r1 = r2 and A = r1 I. Choose the matrix ￿ ￿ ￿ 10 r T= =⇒ T−1 A T = 1 01 0 Case #3: r1 = r2 and A ￿= r1 I. Choose a nonzero vector ￿￿ η η= 1 via the relation η2 Since A η = ξ (1) + r1 η , we set ￿ ￿ ξ η1 T = 11 =⇒ ξ21 η2 AT = ￿ r1 ξ11 r1 ξ21 =⇒ ￿ 0 =J r1 T−1 A T = D. where c = 0. (r1 I − A) η = −ξ (1) . ￿ ξ11 + r1 η1 = TJ ξ21 + r1 η2 =⇒ T−1 A T = J; where c = 1. Since (r1 I − A) ξ (1) = 0, we say that ξ (1) is an eigenvector of A. Similarly, since (r1 I − A) η = −ξ (1) we say that η is a generalized eigenvector of A. This proof shows that when A has repeated eigenvalues, we can always choose T such that c = 0 or 1. Example. Consider the following 2 × 2 matrix: A= ￿ 1 1 ￿ −1 . 3 Recall that we have the eigenvalue r1 = r2 = 2. Define the vectors ￿￿ ￿￿ 1 0 ξ (1) = and η = =⇒ (r1 I − A) ξ (1) = 0 and −1 −1 Hence we choose the 2 × 2 matrix ￿ ￿￿ ξ11 η1 1 T= = ξ21 η2 −1 ￿ 0 −1 =⇒ 1 J=T −1 (r1 I − A) η = −ξ (1) . ￿ 2 AT = 0 ￿ 1 . 2 ...
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