MT Exercise-2-정태일

Exersice 2)

Let $q$ be a quadratic form on $\mathbb{R}^3$ and let  $A=\begin{bmatrix} 7 &4 &-5 \\ 4 &-2 &-4 \\ -5& 4 & 7 \end{bmatrix}$ be the matrix representing $q$ with respect to the basis

$\alpha =\left \{ (1,0,1),(1,1,0),(0,0,1) \right \}$.

1) Diagonalize $A$, i.e., find an orthogonal matrix $P$ so that $P^{-1}AP$ is a diagonal matrix.

2) Construct a basis  $\beta$  for  $\mathbb{R}^3$ such that elements of  $\beta$ are the principal axes of the quadratic surface $q(\rm\bold x)=0$.

Sol)

1)

2) Look at the principal axes theorem.