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lecture_38

# lecture_38 - MA 36600 LECTURE NOTES MONDAY APRIL 27...

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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 r 1 and r 2 . i. If r 1 6 = r 2 i.e., we have distinct eigenvalues, then there exist 2 × 2 matrices T = ξ 11 ξ 12 ξ 21 ξ 22 and D = r 1 r 2 such that T- 1 AT = D . ii. If r 1 = r 2 i.e., we have repeated eigenvalues, then there exist 2 × 2 matrices T = ξ 11 η 1 ξ 21 η 2 and J = r 1 c r 1 such that T- 1 AT = J . Here, either c = 0 or c = 1. In fact, c = 0 if and only if A = r 1 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 r 1 and r 2 as ξ (1) = ξ 11 ξ 21 and ξ (2) = ξ 12 ξ 22 ....
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lecture_38 - MA 36600 LECTURE NOTES MONDAY APRIL 27...

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