Constructing a matrix for a prescribed eigenspace

[morsel]

Homework Statement

Find a 2x2 matrix A for which E1 = span [ 2 1 ] (vertical matrix) is the only eigenspace.

Homework Equations

The Attempt at a Solution

I don't know how to begin this problem.. Any hints?

 

[HallsofIvy]

Do you understand that there are an infinite number of answers? If the "eigenspace" can be spanned by a single vector, then there cannot be two independent eigenvectors- and that means, in turn, that there cannot be two distinct eigenvalues. But since you are not given the single eigenvalue, it can be anything. And if a 2 by 2 matrix has only a one independent eigenvector, it is similar to a "Jordan Normal Form"
$$\begin{bmatrix} a & 1 \\ 0 & a\end{bmatrix}$$
where a is the eigenvalue.

Then you can construct a matrix, P, having the given eigenvector <2, 1>, as first column, and just choose a second row that it easy to find the inverse of P.
Then $P^{-1}AP$ will be a matrix having that only multiples of <2, 1> as eigenvectors. Even after you have done that, you will be able to choosed a to be whatever you want.

 

[morsel]

Yes, I know that there's an infinite number of answers but I was wondering whether there was a structured method for finding one.

Thank you for your explanation!