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\title{MAT 247 \\ Assignment 8 \\ Due Thursday March 25}



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\maketitle
\begin{enumerate}
\item Let $ V, \langle, \rangle $ be a real inner product space.  Let $  T $ be a self-adjoint operator on $ V$.  On the previous assignment, we saw that $ T $ defines a symmetric bilinear form $ H $ by the formula $ H(v,w) = \langle Tv ,w \rangle $.  
\begin{enumerate}
\item Show that $ null(T) = rad(H) $.
\item Find the signature of $ H$ in terms of information about the eigenvalues of $ T $.
\end{enumerate}

\item
Let $ V, W $ be two real vector spaces of the same dimension and let $ H_V, H_W $ be symmetric bilinear forms on $ V, W $ respectively.  We say that an invertible linear map $ T : V \rightarrow W $ is an orthogonal isomorphism if
$$ H_V(v_1, v_2) = H_W(Tv_1, Tv_2), \text{ for all } v_1, v_2 \in V.
$$

Prove that there exists an orthogonal isomorphism $ T : V \rightarrow W $ if and only if the signature of $ H_V $ is the same as the signature of $ H_W $.

\item
Let $ V = \mathbb{R}^2 $.  Define a bilinear form $H_A $ on $ V $ using the matrix
$$
A = \begin{bmatrix} -2 & 1 \\ 1 & -1 \end{bmatrix}.
$$
What is the signature of $ H$?

\end{enumerate}

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