Eigenvalues & Eigenvectors
Linear Algebra
Intro: Compute eigenvalues and eigenvectors step-by-step.
Worked example
- Find eigenvalues and eigenvectors of $A=\begin{pmatrix}4 & 1\\2 & 3\end{pmatrix}$
- Form the characteristic polynomial: $\det(A-\lambda I)=\begin{vmatrix}4-\lambda & 1 \\ 2 & 3-\lambda\end{vmatrix}=(4-\lambda)(3-\lambda)-2 = \lambda^2-7\lambda+10$.
- Solve $\lambda^2-7\lambda+10=0\;\Rightarrow\;(\lambda-5)(\lambda-2)=0$. Thus eigenvalues are $\boxed{\lambda_1=5}$ and $\boxed{\lambda_2=2}$.
- Eigenvector for $\lambda_1=5$: Solve $(A-5I)\mathbf{v}=\mathbf{0}$ with $A-5I=\begin{pmatrix}-1 & 1\\ 2 & -2\end{pmatrix}$. The equation $-x+y=0$ gives $y=x$. Take $\mathbf{v}_1=\begin{pmatrix}1\\1\end{pmatrix}$ (any nonzero multiple works).
- Eigenvector for $\lambda_2=2$: Solve $(A-2I)\mathbf{v}=\mathbf{0}$ with $A-2I=\begin{pmatrix}2 & 1\\ 2 & 1\end{pmatrix}$. The equation $2x+y=0$ gives $y=-2x$. Take $\mathbf{v}_2=\begin{pmatrix}1\\-2\end{pmatrix}$.
- Optional (unit vectors): $\hat{\mathbf{v}}_1=\tfrac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}$, $\hat{\mathbf{v}}_2=\tfrac{1}{\sqrt{5}}\begin{pmatrix}1\\-2\end{pmatrix}$.
- Diagonalization (optional): With $P=[\mathbf{v}_1\;\mathbf{v}_2]=\begin{pmatrix}1&1\\1&-2\end{pmatrix}$ and $D=\operatorname{diag}(5,2)$, we have $A=PDP^{-1}$ since $\det P=-3\ne0$.
FAQs
Complex eigenvalues?
Yes, complex values are supported.
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