(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose V is a complex vector space and [tex]T \in L(V)[/tex]. Prove that

there does not exist a direct sum decomposition of V into two

proper subspaces invariant under T if and only if the minimal

polynomial of T is of the form [tex](z - \lambda)^{dim V}[/tex] for some [tex]\lambda \in C[/tex].

2. Relevant equations

3. The attempt at a solution

First suppose that the minimal polynomial of T is [tex]p(z) = (z - \lambda)^{n}[/tex] where n = dim V. Suppose further that [tex]V = U \oplus W[/tex] where U and W are T-invariant proper subspaces of V. So [tex]dim U \geq 1[/tex] and [tex]dim W \geq 1[/tex]. Since n = dim U + dim W, we have that [tex]dim U \leq n-1[/tex] and [tex]dim W \leq n-1[/tex].

Now the minimal polynomial of T is [tex]p(z) = (z - \lambda)^{n}[/tex], therefore [tex](T - \lambda I)^{n}v = 0[/tex] for all v in V but there is at least one v in V such that [tex](T - \lambda I)^{n-1}v \neq 0[/tex]. Because of the decomposition of V, we can write: v = u + w for some u in U and some w in W. Applying [tex](T - \lambda I)^{n-1} [/tex] to both sides we get:

[tex](T - \lambda I)^{n-1}v = (T - \lambda I)^{n-1}u + (T - \lambda I)^{n-1}w \neq 0[/tex]

since both U and W are invariant under T, we have Tu in U and and [tex] -\lambda u \in U [/tex] so [tex](T - \lambda I)u \in U [/tex]. Thefore U (and similarly W) are both invariant under [tex](T-\lambda I) [/tex], so invariant under [tex](T-\lambda I)^{n-1} [/tex]. [tex](T - \lambda I)^{n-1}v \neq 0 [/tex], so either one of [tex](T - \lambda I)^{n-1}u [/tex] or [tex](T - \lambda I)^{n-1}w [/tex] is not zero, because they're in different subspaces. Assume it's the first one. So there exists u in U such that

[tex](T - \lambda I)^{n-1}u \neq 0[/tex]. But [tex](T - \lambda I)^{n}u = 0 [/tex] (for all u in U). Constraining T to U, we see that the minimal polynomial of T on U is [tex](z - \lambda I)^{n} [/tex]. But [tex] dim U \leq n-1 < n [/tex], a contradiction.

I hope the reasoning above is correct; if it is, it proves that minimal polyomial of T is [tex](z-\lambda I)^{n} [/tex] implies that T cannot be decomposed into the direct sum of two proper T-invariant subspaces.

However I'm stumped about how to prove the other direction.

Also, I would like to know what's the mistake in the following reasoning:

Since V is a complex vector space, then T in L(V) has an eigenvalue [tex]\lambda[/tex], so V has an T-invariant subspace of dimension 1. Then there exists a subspace W of V such that [tex]V = U \oplus W[/tex]. It follows that W is T-invariant and of dimension dim V - 1[/tex]. So it's always possible to decompose V into two proper T-invariant proper subspaces :S

Thank you :)

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# Homework Help: Decomposition of a complex vector space into 2 T-invariant subspaces

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