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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