Linear Mathematics | Part IB, 2001

(a) Find a matrix MM over C\mathbb{C} with both minimal polynomial and characteristic polynomial equal to (x2)3(x+1)2(x-2)^{3}(x+1)^{2}. Furthermore find two matrices M1M_{1} and M2M_{2} over C\mathbb{C} which have the same characteristic polynomial, (x3)5(x1)2(x-3)^{5}(x-1)^{2}, and the same minimal polynomial, (x3)2(x1)2(x-3)^{2}(x-1)^{2}, but which are not conjugate to one another. Is it possible to find a third such matrix, M3M_{3}, neither conjugate to M1M_{1} nor to M2M_{2} ? Justify your answer.

(b) Suppose AA is an n×nn \times n matrix over R\mathbb{R} which has minimal polynomial of the form (xλ1)(xλ2)\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right) for distinct roots λ1λ2\lambda_{1} \neq \lambda_{2} in R\mathbb{R}. Show that the vector space V=RnV=\mathbb{R}^{n} on which AA defines an endomorphism α:VV\alpha: V \rightarrow V decomposes as a direct sum into V=ker(αλ1ι)ker(αλ2ι)V=\operatorname{ker}\left(\alpha-\lambda_{1} \iota\right) \oplus \operatorname{ker}\left(\alpha-\lambda_{2} \iota\right), where ι\iota is the identity.

[Hint: Express vVv \in V in terms of (αλ1ι)(v)\left(\alpha-\lambda_{1} \iota\right)(v) and (αλ2ι)(v).]\left.\left(\alpha-\lambda_{2} \iota\right)(v) .\right]

Now suppose that AA has minimal polynomial (xλ1)(xλ2)(xλm)\left(x-\lambda_{1}\right)\left(x-\lambda_{2}\right) \ldots\left(x-\lambda_{m}\right) for distinct λ1,,λmR\lambda_{1}, \ldots, \lambda_{m} \in \mathbb{R}. By induction or otherwise show that

V=ker(αλ1ι)ker(αλ2ι)ker(αλmι)V=\operatorname{ker}\left(\alpha-\lambda_{1} \iota\right) \oplus \operatorname{ker}\left(\alpha-\lambda_{2} \iota\right) \oplus \ldots \oplus \operatorname{ker}\left(\alpha-\lambda_{m} \iota\right)

Use this last statement to prove that an arbitrary matrix AMn×n(R)A \in M_{n \times n}(\mathbb{R}) is diagonalizable if and only if all roots of its minimal polynomial lie in R\mathbb{R} and have multiplicity 1.1 .

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