Hello,

I have a question to do which I'm a bit stuck on. Any and all help would be much appreciated!

Assume V is an inner product space. If dim(V) = 3, for which of the following matrices does there exist a self-adjoint linear transformation $\displaystyle T_{i}$ on V for which the matrix $\displaystyle A_{i}$ represents $\displaystyle T_{i}$ with respect to some basis?

$\displaystyle A_{0}=\begin{pmatrix}

2 & 0 & 0\\

0 & 1 & 1\\

0 & 0 & 0

\end{pmatrix}

A_{1}=\begin{pmatrix}

2 & 0 & 0\\

0 & 1 & 1\\

0 & 0 & 1

\end{pmatrix}

A_{2}=\begin{pmatrix}

2 & 0 & 0\\

0 & 1 & 1\\

0 & 0 & 2

\end{pmatrix}$

For the first one I'm thinking there's not, because of the row of 0's at the bottom. However, I'm still not 100% sure.

(I understand what a self-adjoint linear transformation is, and I also know that the eigenvectors of V will form an orthonormal basis of V.)