+25 I was wondering whether there is a way of solving these questions without calculating A.
I figured they probably didn't 'split-up' matrix A into three matrices for no reason.
This is the question;
$$\mbox{Consider A = } \begin{pmatrix}
-3 & -2 & 0\\
-1 & -1 & 0\\
0 & 0 & 1\\
\end{pmatrix}
\begin{pmatrix}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1\\
\end{pmatrix}
\begin{pmatrix}
-1 & 2 & 0\\
1 & -3 & 0\\
0 & 0 & 1
\end{pmatrix}$$
a) Is A similar to a diagonal matrix? Explain the answer.
I was able to solve this one, it is obvious since $$S = \begin{pmatrix}
-1 & 2 & 0\\
1 & -3 & 0\\
0 & 0 & 1
\end{pmatrix}, B = \begin{pmatrix}
1 & 0 & 0\\
0 & 0 & 0\\
0 & 0 & 1\\
\end{pmatrix}, A = S^{-1}BS$$
b) Is A idempotent? Explain the answer in two different ways.
c) Is A (semi)-definite? Explain the answer.
d) Is A congruent to a diagonal matrix? Explain the answer.
e) Is A orthogonal equivalent to a diagonal matrix? Explain the answer.
