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THE GEOMETRY OF MATHEMATICAL METHODS

Section 4.2 Finding Eigenvalues

Method for Finding Eigenvalues of a Square Matrix.

In order to find the eigenvalues of a square matrix \(A\text{,}\) we must find the values of \(\lambda\) such that the equation
\begin{equation} A \left|v\right> = \lambda \left|v\right>\tag{4.2.1} \end{equation}
admits solutions \(\left|v\right>\text{.}\) (The solutions \(\left|v\right>\) are eigenvectors of \(A\text{,}\) as discussed in the next section.) Rearranging terms, \(\left|v\right>\) must satisfy
\begin{equation} (\lambda I-A)\left|v\right> = 0\tag{4.2.2} \end{equation}
where \(I\) denotes the identity matrix (of the same size as \(A\)). Suppose that the inverse matrix \((\lambda I-A)^{-1}\) exists. Multiplying both sides of (4.2.2) by this inverse would then yield
\begin{equation} (\lambda I-A)^{-1}(\lambda I-A)\left|v\right> = \left|v\right> = 0\tag{4.2.3} \end{equation}
which is not a very interesting solution. So, instead, we want \((\lambda I-A)\) not to have an inverse. When does this happen?
Claim: A square matrix \(A\) does not have an inverse if and only if \(\det A=0\text{.}\)

Definition 4.3. Degeneracy/Multiplicity.

In the case of an eigenvalue that is repeated \(m\) times, we call the eigenvalue \(\boldsymbol{m}\)-fold degenerate, or, equivalently, of multiplicity \(\boldsymbol{m}\). For more information about degenerate eigenvalues, see Section 4.6.

Example.

Suppose \(A=\begin{pmatrix}1\amp 2\\9\amp 4 \end{pmatrix}\text{.}\) Then we must solve
\begin{equation} 0 = |\lambda I-A| = \begin{vmatrix}1\amp 2\\9\amp 4 \end{vmatrix} = (\lambda-1)(\lambda-4)-18\tag{4.2.4} \end{equation}
or equivalently
\begin{equation} 0 = \lambda^2 - 5\lambda -14 = (\lambda-7)(\lambda+2)\text{,}\tag{4.2.5} \end{equation}
and the eigenvalues in this case are \(\lambda=7\) and \(\lambda=-2\text{.}\)