In this chapter we look at the diagonalization problem for real symmetric matrices. You probably saw how to compute eigenvalues and eigenvectors in your elementary linear algebra course. You may have also seen that in some cases, the number of independent eigenvectors associated to an \(n\times n\) matrix \(A\) is \(n\text{,}\) in which case it is possible to “diagonalize” \(A\text{.}\) In other cases, we don’t get “enough” eigenvectors for diagonalization.
In the first part of this section, we review some basic facts about eigenvalues and eigenvectors. We will then move on to look at the special case of symmetric matrices, where we will see that it is always possible to diagonalize, and moreover, that it is possible to do so using an orthonormal basis of eigenvectors.
