1314 denition for a mn the characteristic polynomial

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Unformatted text preview: s called the spectrum of A. 1.3.13 Remark. λ is an eigenvalue ⇔ ∃x ￿= 0, Ax = λx. • If we write the last portion as (λI − A)x = 0, we can see that (λI − A) is not invertible and therefore, det(λI − A) = 0 (Note that the latter relation is also necessary and sufficient). 1.3.14 Definition. For A ∈ Mn , the characteristic polynomial is defined by: PA (t) = det(tI − A) 1.3.15 Remark. With the insight from above, λ is an eigenvalue ⇔ PA (λ) = 0 • Over C, PA can be factorized so that, PA (t) = (t − λ1 )(t − λ2 ) . . . (t − λn ) where some λj s may be repeated and 1 ≤ j ≤ n. Comparing the definition of PA with the factorized form gives, n PA (t) = t + (−1) n ￿ λj t j =1 3 n −1 + · · · + (−1) n n ￿ j =1 λj From the definition of the determinant, we see that det(tI − A) = tn − tr[A]tn−1 + · · · + (−1)n det(A) By comparing terms in the polynomial, we also notice that tr[A] = n ￿ λj j =1 and det[A] = n ￿ λj j =1 1.4 Similarity 1.4.16 Definition. A matrix B ∈ Mn is similar to A ∈ Mn...
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This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.

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