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Unformatted text preview: if ∃S ∈ Mn , where S is invertible,
B = S −1 AS
This deﬁnes an equivalence relation, as can be veriﬁed.
1.4.17 Theorem. If A, B ∈ Mn are similar, then PA = PB
PB (t) = det(tI − B )
= det(t(S −1 S ) − S −1 AS )
= det(S −1 (tI − A)S )
= det(S −1 )det(tI − A)det(S )
= det(S −1 S )det(tI − A)
= det(tI − A) = PA (t)
We conclude A, B share same eigenvalues with the same multiplicity.
1.4.18 Remark. No...
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