Eigenvalues and Eigenvectors

Eigenbasis

Is a specific basis of a vector space in which every vector is an eigenvector of a given linear transformation. The eigenvectors provides the direction of a stretch on the transformation space while the eigenvalues gives the magnitude of this stretch.

Calculating Eigenvalues and Eigenvectors

• $v_1$ and $v_2$ are the eigenvectors
• ${\lambda }_1$ and ${\lambda }_2$ are the eigenvalues

$$\begin{array}{r}\mathbf{A}{v}_1={\lambda }_1{v}_1\\ \mathbf{A}{v}_2={\lambda }_2{v}_2\end{array}$$

• $\mathbf{A}v=\lambda v$ for each eigenvector / eigenvalue
• Eigenvectors: direction of stretch
• Eigenvalues: how much stretch
• Eigenbasis: the set of a matrix’s eigenvectors, can be arranged as a matrix with one eigenvector in each column
• Save work and characterize a transformation

Finding Eigenvalues

If $\lambda$ is an eigenvalue, for infinitely many $(x,y)$:

$$\left[\begin{array}{cc}2& 1\\ 0& 3\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{cc}\lambda & 0\\ 0& \lambda \end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]$$ $$\left[\begin{array}{cc}2-\lambda & 1\\ 0& 3-\lambda \end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}0\\ 0\end{array}\right]$$

Has infinitely many solutions.

$$det\left[\begin{array}{cc}2-\lambda & 1\\ 0& 3-\lambda \end{array}\right]=0$$

Characteristic polynomial:

$$(2-\lambda )(3-\lambda )-1\cdot 0=0$$

$\lambda =2$ and $\lambda =3$.

General formula:

$$\boxed{det(\mathbf{A}-\lambda \mathbf{I})=0}$$

Finding Eigenvectors

Solve the equations:

$$\left[\begin{array}{cc}2& 1\\ 0& 3\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]=2\cdot \left[\begin{array}{c}x\\ y\end{array}\right]$$ $$\left[\begin{array}{cc}2& 1\\ 0& 3\end{array}\right]\cdot \left[\begin{array}{c}x\\ y\end{array}\right]=3\cdot \left[\begin{array}{c}x\\ y\end{array}\right]$$