Definition 4.6. Eigenspace.
An eigenspace is the set of all vectors with the same eigenvalue.
Section 4.6 Degeneracy and Eigenspaces
An Example of Degenerate Eigenvalues.
It is not always the case that an \(n\times n\) matrix has \(n\) distinct eigenvectors. In Section 4.2, we defined a repeated eigenvalue to be degenerate. For example, consider the matrix
\begin{equation}
B =
\begin{pmatrix}
3 \amp 0 \amp 0\\
0 \amp 3 \amp 0\\
0 \amp 0 \amp 5
\end{pmatrix}\text{,}\tag{4.6.1}
\end{equation}
whose eigenvectors are clearly the standard basis. But what are the eigenvalues of \(B\text{?}\) Again, the answer is obvious: \(3\) and \(5\text{.}\) In such cases, the eigenvalue \(3\) is a degenerate eigenvalue of \(B\text{,}\) since there are two linearly independent eigenvectors of \(B\) with eigenvalue \(3\text{.}\) In this case, one also says that \(3\) is a repeated eigenvalue of multiplicity \(\boldsymbol{2}\).
However, that’s not the whole story. Setting
\begin{equation}
|v\rangle \doteq \begin{pmatrix}1\\0\\0
\end{pmatrix},
\qquad
|w\rangle \doteq \begin{pmatrix}0\\1\\0
\end{pmatrix},\tag{4.6.2}
\end{equation}
we of course have
\begin{equation}
B|v\rangle = 3|v\rangle, \qquad B|w\rangle = 3|w\rangle\text{.}\tag{4.6.3}
\end{equation}
But, by the linearity of matrix operations (see Section 3.2 and Section 3.1), we also have
\begin{equation}
B(a|v\rangle + b|w\rangle) = a\,B|v\rangle + b\,B|w\rangle
= 3(a|v\rangle + b|w\rangle)\text{,}\tag{4.6.4}
\end{equation}
so that any linear combination of \(|v\rangle\) and \(|w\rangle\) is also an eigenvector of \(B\) with eigenvalue \(3\text{.}\)
The eigenspace of \(B\) corresponding to eigenvalue \(3\) is therefore a 2-dimensional vector space (a plane), rather than the 1-dimensional eigenspaces (lines) that occur when the eigenvalues are distinct.
