Section 5.8 Diagonalization: Using Eigenvectors as a Natural Basis
The Eigenvectors of Hermitian Matrices form a Natural Basis.
In
Section 5.3, we showed that the eigenvectors of Hermitian (and anti-Hermitian) matrices are orthogonal, or in the case of degeneracy, can be chosen to be orthogonal. In
Section 5.5, we showed the same for unitary (and anti-unitary) matrices.
A much deeper question is whether an \(n\times n\) matrix actually has a full \(n\) eigenvectors. The spectral theorem guarantees that this is the case for Hermitian, anti-Hermitian, unitary, and anti-unitary matrices, but the proof is well beyond the scope of this book. The technical language for this is that the eigenvectors of these matrices form a complete set, i.e. they span the entire vector space.
Since eigenvectors can always be normalized, see
Section 3.5, we see that the eigenvectors of these special matrices form an orthonormal basis.
The next activity explores one of the important implications of using the eigenvectors as a natural basis.
Activity 5.3. Find the Form of a Hermitian Operator in Its Own Eigen-Basis.
Consider the Hermitian operator
\(A\text{.}\) In a coordinate system where its eigenvectors are the standard basis
(4.5.4), name its components
\begin{equation}
A\doteq\begin{pmatrix}
a_{11} \amp a_{12}\\ a_{21} \amp a_{22}
\end{pmatrix}\text{.}\tag{5.8.1}
\end{equation}
Show that the eigenvalue/eigenvector equation
(4.1.1) requires that the diagonal components are the eigenvalues and the off-diagonal elements are zero.
Hint.
One of the eigenvalue/eigenvector equations states that
\begin{equation}
\begin{pmatrix}
a_{11} \amp a_{12}\\ a_{21} \amp a_{22}
\end{pmatrix}
\begin{pmatrix} 1 \amp 0 \end{pmatrix}
= \lambda_1 \begin{pmatrix} 1 \amp 0 \end{pmatrix}\text{.}\tag{5.8.2}
\end{equation}
Compare the left- and right-hand sides of this equation.
In the activity above, you showed that, for the simplest example of a \(2\times 2\) matrix, in a coordinate system where the eigenvectors of a Hermitian matrix are the standard basis, the Hermitian matrix is diagonal and its diagonal elements are just the eigenvalues. This result is true, in general, for any size matrix that is Hermitian, anti-Hermitian, unitary, or anti-unitary.
The process of diagonalization is the process of changing basis to the natural eigen-basis of such an operator. To actually find the change-of-basis operation requires some algebra (which we will not discuss further here). But scientists often casually say that they are ``diagonalizing’’ a matrix when all they really do is state the diagonal matrix that is the result.
To Remember.
A Hermitian matrix is diagonal in its own eigen-basis and, in this coordinate system, its eigenvectors are just the standard basis
(4.5.4). Many calculations are simplified by using this basis.
