Eigenvalues and Eigen vectors

What are Eigenvalues and eigen vectors

• Recall in the (not yet noted) the standard matrix for the reflection about the line y =2x:

$$\left[\begin{array}{c}L\end{array}\right]=\left[\begin{array}{cc}-\frac{3}{5}& \frac{4}{5}\\ \frac{4}{5}& \frac{3}{5}\end{array}\right]$$

• And when we scale a vector by this the vector it self changes

• An eigenvalue/vector is a matrix that was part of the original matrix that stays the same after the transformation

  • let A be an nxn matrix. if there exists a nonzero vector $\vec{v}\in {\mathbb{R}}^n$ such that $A\vec{v}=\lambda \vec{v}$ for some $\lambda \in \mathbb{R}$

    • we say $\lambda$ is an eigen value of A and $\vec{v}$ is an eigenvector of A corresponding to $\lambda$

How to find eigenvalues and eigenvectors

• Eigenvalues

  • Suppose $\vec{v}$ is a nonzero vector such that $A\vec{v}=\lambda \vec{v}$ then
    • $A\vec{v}=\lambda \vec{v}=(\lambda I_n)\vec{v}⟹A\vec{v}-(\lambda I_n)\vec{v}=0⟹(A-\lambda I_n)\vec{v}=\vec{0}$
      • that is, $\vec{v}$ is a nontrivial solution to $(A-\lambda I)\vec{x}=\vec{0}$
        • for $\vec{v}$ to be non zero it $(A-\lambda I)$ Determinant needs to be non zero

• Eigenvectors

  • once we got the eigenvalues of A are found, we can find all eigen vectors (the vectors that dont change under the transformation) by solving $(A-\lambda I)\vec{x}=\vec{0}$
    • so on the diagonals you would remove $\lambda$ and find the resulting matrix (like finding the linear combination for the zero vector)

• Characteristic polynomial!

  • the eigen polynomail of the vector (remember)