Paper 1, Section II, 7B

Vectors and Matrices | Part IA, 2019

(a) Let AA be an n×nn \times n matrix. Define the characteristic polynomial χA(z)\chi_{A}(z) of AA. [Choose a sign convention such that the coefficient of znz^{n} in the polynomial is equal to (1)n.]\left.(-1)^{n} .\right] State and justify the relation between the characteristic polynomial and the eigenvalues of AA. Why does AA have at least one eigenvalue?

(b) Assume that AA has nn distinct eigenvalues. Show that χA(A)=0\chi_{A}(A)=0. [Each term crzrc_{r} z^{r} in χA(z)\chi_{A}(z) corresponds to a term crArc_{r} A^{r} in χA(A).]\left.\chi_{A}(A) .\right]

(c) For a general n×nn \times n matrix BB and integer m1m \geqslant 1, show that χBm(zm)=l=1mχB(ωlz)\chi_{B^{m}}\left(z^{m}\right)=\prod_{l=1}^{m} \chi_{B}\left(\omega_{l} z\right), where ωl=e2πil/m,(l=1,,m).[\omega_{l}=e^{2 \pi i l / m},(l=1, \ldots, m) .[ Hint: You may find it helpful to note the factorization of zm1z^{m}-1.]

Prove that if BmB^{m} has an eigenvalue λ\lambda then BB has an eigenvalue μ\mu where μm=λ\mu^{m}=\lambda.

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