Eigenvalues and Eigenvectors

The existence of an Eigenvector

Let $\mathit{A}$ be an $\mathit{n}\times \mathit{n}$ matrix and suppose $\det \left(\mathit{\lambda I}-\mathit{A}\right)=0$ for some $\mathit{\lambda }\in \mathit{ℂ}$.
Then $\mathit{\lambda }$ is an eigenvalue of $\mathit{A}$ and thus there exists a nonzero vector $\mathit{X}\in {\mathit{ℂ}}^{\mathit{n}}$ such that $\mathit{AX}=\mathit{\lambda X}$.

The expression $\det \left(\mathit{\lambda I}-\mathit{A}\right)$ is a polynomial (in the variable $\mathit{x}$) called the characteristic polynomial of $\mathit{A}$, and $\det \left(\mathit{\lambda I}-\mathit{A}\right)=0$ is called the characteristic equation. For this reason, we may also refer to the eigenvalues of $\mathit{A}$ as characteristic values.

Multiplicity of an Eigenvalue

Let $\mathit{A}$ be an $\mathit{n}\times \mathit{n}$ matrix with characteristic polynomial given by $\det \left(\mathit{\lambda I}-\mathit{A}\right)$. Then, the multiplicity of an eigenvalue $\mathit{\lambda }$ of $\mathit{A}$ is the number of times $\mathit{\lambda }$ occurs as a root of that characteristic polynomial.

For example, suppose the characteristic polynomial of $\mathit{A}$ is given by ${\left(\mathit{\lambda }-2\right)}^2$. Solving for the roots of this polynomial, we set ${\left(\mathit{\lambda }-2\right)}^2=0$ and solve for $\mathit{\lambda }$. We find that $\mathit{\lambda }=2$ is a root that occurs twice. Hence, in this case, $\mathit{\lambda }=2$ is an eigenvalue of $\mathit{A}$ of multiplicity equal to $2$.

Finding Eigenvalues and Eigenvectors

Let $\mathit{A}$ be an $\mathit{n}\times \mathit{n}$ matrix.

1. First, find the eigenvalues $\mathit{\lambda }$ of $\mathit{A}$ by solving the equation $\det \left(\mathit{\lambda I}-\mathit{A}\right)=0$
2. For each $\mathit{\lambda }$, find the basic eigenvectors $\mathit{X}\ne 0$ by finding the basic solutions to $\left(\mathit{\lambda I}-\mathit{A}\right)\mathit{X}=0$.

To verify your work, make sure that $\mathit{AX}=\mathit{\lambda X}$ for each $\mathit{\lambda }$ and associated eigenvector $\mathit{X}$.