Question
Find an orthogonal basis of eigenvectors for the following matrix. The matrix has a repeated eigenvalue so you will need to use the Gram-Schmidt process.
$$\begin{bmatrix}5 & 4 & 2\\ 4 & 5 & 2 \\ 2 & 2 & 2 \end{bmatrix}$$
($\lambda = 1$ is a double eigenvalue.)
Answer
Well here's what I found for eigenvalues and eigenvectors -
For $\lambda = 1$ I found eigenvectors $\begin{bmatrix}-1 \\ 1 \\ 0 \end{bmatrix}$, $\begin{bmatrix}-1 \\ 0 \\ 2 \end{bmatrix}$
For $\lambda = 10$ I found eigenvector $\begin{bmatrix}2 \\ 2 \\ 1 \end{bmatrix}$
Now I can see that the eigenvectors of $\lambda = 1$ are not orthogonal to each other. But how do I do the rest of the question? It says to use the Gram-Schmidt process but if I use that on one of the vectors it will change it's orientation and it will no longer be an eigenvector. So how do I get an orthogonal basis while getting to keep the basis as eigenvectors?