My question is a prerequisite for If two eigenvectors share eigenvalues, why is there no unique decomposition?
How would I prove that:
While any real symmetric matrix A is guaranteed to have an eigendecomposition, the eigendecomposition may not be unique.
I understand why it may not be unique, but what property guarantees that there is at least one eigendecomposition?
If this statement is equal to "characteristic polynomial of every real symmetrical matrix has a at least one real root", then my question, again, is what property guarantees that?