Paper 1, Section II, B

Vectors and Matrices | Part IA, 2010

Let AA be a complex n×nn \times n matrix with an eigenvalue λ\lambda. Show directly from the definitions that:

(i) ArA^{r} has an eigenvalue λr\lambda^{r} for any integer r1r \geqslant 1; and

(ii) if AA is invertible then λ0\lambda \neq 0 and A1A^{-1} has an eigenvalue λ1\lambda^{-1}.

For any complex n×nn \times n matrix AA, let χA(t)=det(AtI)\chi_{A}(t)=\operatorname{det}(A-t I). Using standard properties of determinants, show that:

(iii) χA2(t2)=χA(t)χA(t)\chi_{A^{2}}\left(t^{2}\right)=\chi_{A}(t) \chi_{A}(-t); and

(iv) if AA is invertible,

χA1(t)=(detA)1(1)ntnχA(t1)\chi_{A^{-1}}(t)=(\operatorname{det} A)^{-1}(-1)^{n} t^{n} \chi_{A}\left(t^{-1}\right)

Explain, including justifications, the relationship between the eigenvalues of AA and the polynomial χA(t)\chi_{A}(t).

If A4A^{4} has an eigenvalue μ\mu, does it follow that AA has an eigenvalue λ\lambda with λ4=μ\lambda^{4}=\mu ? Give a proof or counterexample.

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