The given task is to find all 2×2 Matrices A
that are similar to:
a) $\begin{bmatrix}
0 & 0 \\
0 & 0
\end{bmatrix}$
b) $\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}$
c) $\begin{bmatrix}
1 & 0 \\
0 & 2
\end{bmatrix}$
However, I don't really know which steps to take. Obviously, since they are all in their diagonal form we can see that they have the eigenvalues
a) $\lambda = 0$
b) $\lambda = 1$
c) $\lambda_{\mathrm{1}} = 1,\lambda_{\mathrm{2}} = 2$
Now how do I go about finding the general matrix $A$ that is similar to these? Do I have to define
$A = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$
and calculate its eigenvalues and eigenvectors?
I know that the definition of similar matrices $A,B$ is that there exists some matrix $S$ so that:
$A = S^{-1}BS$
but I don't think that defining a general $S= \begin{bmatrix}
e & f \\
g & h
\end{bmatrix}$
and solving the equation is the proper way to do it. I also have a feeling that this has something to do with Jordan-Blocks but I can't seem to figure it out.
I'm sorry if this is a stupid question or something really simple, but we only had lectures on the whole topics of eigenvalues, eigenvectors and general eigenvectors two days ago.
Best Answer
The first two diagonal matrices$~D$ commute with every matrix, so $S^{-1}DS=SS^{-1}D=D$ for any $S$, so these two are each only similar to themselves. So concentrate on the final matrix$~B$.
You can in principle compute the set $\{\, S^{-1}BS\mid s\in GL(2,\Bbb R)\,\}$ explicitly by using the formula for the inverse and matrix multiplication. You can maybe simplify the task by limiting $S$ to matrices with determinant $1$ or $-1$, since all invertible matrices can be brought into this form by a scalar multiplication, which scalar will not affect $S^{-1}BS$. Then you may find an equation to describe the resulting set.
Since $B$ has all its (two) eigenvalues distinct, there is an easier approach to get the result. Every matrix$~A$ with the same characteristic polynomial $X^2-3X+2$ as $B$ will have the same eigenvalues, and since they are all distinct $A$ will be diagonalisable. This means $A$ is similar to a diagonal matrix, which must be $B$ (possibly with the diagonal entries permuted, but all such diagonal matrices are similar, so being similar to one means being similar to$~B$). So you are looking for the set of matrices with characteristic polynomial $X^2-3X+2$. That set is $$ \left\{\pmatrix{a&b\\c&d}\mid a+d=3,~ad-bc=2 \right\}. $$