I know that different eigenvectors from different eigenspace are automatically orthogonal.
My question is:
Suppose we are doing spectrum decomposition to a 3x3 symmetric matrix and we have only two different eigenvalues(which means we have one repeated eigenvalue). Then we can choose two eigenvectors for that repeated eigenvalue and then do Grand-Schmit orthogonalization to these two vectors. But G-S orthogonalization requires linear independent vector here.
Does it mean:
1.We should always pick well selected linearly independent eigenvectors?
2.if 1 holds, can we always guarantee that we can find linearly independent eigenvectors for repeated eigenvalue?