If $$
A=\begin{bmatrix}
1 & 2 & 0\
2 & -1& -2\
0 & -2& 1
\end{bmatrix}
$$
And it is given that $$P^{T}AP $$ equals to a diagonal matrix, and that P is an orthogonal matrix, how can I find the matrix P?

Question

If $$
A=\begin{bmatrix}
1 & 2 & 0\ 
2 & -1& -2\ 
0 & -2& 1
\end{bmatrix}
$$
And it is given that $$P^{T}AP $$ equals to a diagonal matrix, and that P is an orthogonal matrix, how can I find the matrix P?

zarkon · Answer

find the eigenspace

anonymous · Answer

but the eigenvalues include complex numbers.

zarkon · Answer

-3,1,3 are the eigenvalues

zarkon · Answer

http://alturl.com/c5aiw

anonymous · Answer

ohhh...thanks! i had a calculation mistake. thanks a lot!! :)

zarkon · Answer

np

anonymous · Answer

but I still have a problem... because the question wanted a transpose of P, after finding the eigenspace, to diagonalise the eigenvalue matrix, I need to get the inverse of the eigenspace matrix. If I get an inverse of it, then it isn't a transpose anymore.

zarkon · Answer

did you normalize the eigenvectors?

anonymous · Answer

but why do i have to normalise them?

zarkon · Answer

P is really just supposed to be a rotation matrix (an orthonormal matrix)

anonymous · Answer

wow... can I generalise it to say for all othonormal matrix that belongs to the eigenspace, the transpose is the same as the inverse?

zarkon · Answer

If you do not normalize then you will still diagonalize a but $$p^t\neq p^{-1}$$

anonymous · Answer

Can I generalise it to say all normalised othogonal matrix, its transpose is the same as its inverse?

zarkon · Answer

yes

zarkon · Answer

it is really easy to prove

since $$p^tp=I$$

thus $$p^t=p^{-1}$$

zarkon · Answer

some take the definition of orthogonal matrix to be  a square matrix A such that

$$A^tA=I$$

you should check your book to see how it defines it

anonymous · Answer

oh...thanks a lot for your help! thanks!

zarkon · Answer

np :)