# A linear map $T: \mathbb{R^3 \to \mathbb{R^3}}$ has a two dimensional invariant subspace.

Let $$T: \mathbb{R^3 \to \mathbb{R^3}}$$ be an $$\mathbb{R}$$-linear map. Then I want to show that $$T$$ has a $$2$$ dimensional invariant subspace of $$\mathbb{R^3}.$$

I considered all possible minimal polynomial of $$T$$ and applying canonical forms I found some obvious $$2$$ dimensional invariant subspaces.

I stuck when the minimal polynomial is of the form $$(X-a)^3$$ for some real number $$a.$$

In this situation since the minimal polynomial and the characteristic polynomial coincides $$T$$ has a cyclic vector. But I can't complete it further. I need some help. Thanks.

## 1 Answer

If the minimal polynomial is of the form $$(X-a)^3$$, then there is some basis $$B=\{e_1,e_2,e_3\}$$ of $$\mathbb{R}^3$$ such that the matrix of $$T$$ with respect to $$B$$ is$$\begin{bmatrix}a&1&0\\0&a&1\\0&0&a\end{bmatrix}.$$So, consider the space spanned by $$e_1$$ and $$e_2$$.

• How can it be a diagonal matrix ? also second matrix how do you get ? – user371231 Jan 10 at 11:41
• I wrote “minimal polynomial” but I was thinking about characteristic polynomials. I hope that everything is correct now. – José Carlos Santos Jan 10 at 11:43