Describe all matrices similar to a certain matrix. Math people:
I assigned this problem as homework to my students (from Strang's "Linear Algebra and its Applications", 4th edition): 

Describe in words all matrices that are similar to $$\begin{bmatrix}1& 0\\ 0& -1\end{bmatrix}$$ and find two of them.

Square matrices $A$ and $B$ are defined to be "similar" if there exists square invertible $M$ with $A = M^{-1}BM$ (or vice versa, since this is an equivalence relation).  The answer to the problem is not in the text, and I am embarrassed to admit I am having trouble solving it.  The problem looked easy when I first saw it.
The given matrix induces a reflection in the $x_2$-coordinate, but I don't see how the geometry helps.  A similar matrix has to have the same eigenvalues, trace, and determinant, so its trace is $0$ and its determinant is $-1$.  I spent a fair amount of time on it, with little progress, and I can spend my time more productively.  This problem is #2 in the problem set, which suggests that maybe there is an easy solution.
I would settle for a hint that leads me to a solution.    
EDIT: Thanks to Thomas (?) for rendering my matrix in $\LaTeX$.
Stefan (STack Exchange FAN)

The question says "describe in words", so I think an answer without much formulas is in place. Conjugates of this matrix will be precisely those $2\times2$ matrices with eigenvalues $+1$ and $-1$, in other words those which fix the scalar multiples of one nonzero vector, and negates the scalar multiples of another (necessarily linearly independent) vector. Another description is any matrix of an involution (i.e., with square the identity) other that plus or minus the identity (since their minimal polynomial $X^2-1$ is split with simple roots, involutions are always diagonalisable (except in characteristic $2$)).
