How do you orthogonally diagonalize the following matrix:
$\displaystyle
A=\begin{pmatrix}1&0&0\\0&3&-2\\0&-2&3\end{pmatrix}
$
hi
You need to find the eigenvalues of the matrix, and find the corresponding eigenvectors. Do you know how to do this?
$\displaystyle A = PDP^{T} $ , since A is orthogonally diagonizable, P will be an orthogonal matrix, hence $\displaystyle P^{-1}=P^{T} $ .
An easy way to find the eigenvalues of this matrix would be a cofactor expansion for example, or you could use the fact that it is block triangular.
Itīs quite a long process to describe if you have no idea whatsoever on how to find eigenvalues or eigenvectors, and I would in that case suggest you start there.
hi
Yes, the eigenvalus are $\displaystyle \lambda_{1} = 5 \; \lambda_{2} = 1 \; \lambda_{3} = 1 $ so the multiplicity of eigenvalue 1 is two. This means that the dimension of the nullspace of the matrix $\displaystyle (A- 1 \cdot I) $ has dimension 2.
The eigenvectors you wrote are correct.
There is a theorem that says that eigenvectors from different eigenspaces to a symmetric matrix are orthogonal, so you only need to verify that the two vectors corresponding to eigenvalue 1 are orthogonal to each other.
Which they are here.
Now you normalize your eigenvectors and put them in a matrix P, and the eigenvalues you put in a diagonal matrix D, with the eigenvalue corresponding to the first column vector in the first column of D etc. Order matters here!
And you are done!