Im really lost on how to do these, i got no idea, can someone explain what i need to do to solve this, thanks
Orthogonally diagonalize the symmetric matrix:
0 1 0
1 0 0
0 0 3
I have no idea what the definition of "orthogonal diagonalise" is. To diagonalise this matrix $\displaystyle S$, you need to swap the role of the first and second column and keep third fixed, so use $\displaystyle P$
0 1 0
1 0 0
0 0 1
and you will see that $\displaystyle PSP^{-1}$ is a diagonal matrix. In case "orthogonal diagonalise" means $\displaystyle PP^t=I$, then this $\displaystyle P$ should work.