Eigenvectors & Eigenvalues

91. Eigenvectors & Eigenvalues

91.1. Eigen

𝐴 𝑥 = 𝜆 𝑥

When a transformation $𝐴$ is applied to a vector $𝐱$ , it is equivalent to scaling the vector by a factor of $𝜆$ . The vector $𝐱$ is called an eigenvector of the transformation $𝐴$ , and the scalar $𝜆$ is called the corresponding eigenvalue.

Example

Transformation

Cectors $\hat{𝑖}, \hat{𝑗}, \vec{𝑥}$ under transformation $𝐴$

𝐴 = [\begin{matrix} 3 & 1 \\ 0 & 2 \end{matrix}]

Example

Transformation

Vectors $\hat{𝑖}, \hat{𝑗}, {\hat{𝑣}}_{1}, {\hat{𝑣}}_{2}$ under transformation $𝐴$

𝐴 = [\begin{matrix} 2 & 1 \\ 1 & 2 \end{matrix}]

An $𝑛 \times 𝑛$ matrix $𝐴$ has exactly $𝑛$ eigenvalues because the eigenvalues are the roots of its characteristic polynomial, which is degree $𝑛$

Characteristic polynomial:

𝑝 (𝜆) = det (𝐴 - 𝜆 𝐼)

Example

$2 \times 2$ Matrix

𝐴 = [\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}]

det (𝐴 - 𝜆 𝐼) = (2 - 𝜆) (3 - 𝜆) = 0

Solve for $𝜆$ :

𝜆_{1} = 2, 𝜆_{2} = 3

$3 \times 3$ Matrix

𝐴 = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 4 \end{matrix}]

det (𝐴 - 𝜆 𝐼) = (1 - 𝜆) (2 - 𝜆) (4 - 𝜆) = 0

Solve for $𝜆$ :

𝜆_{1} = 1, 𝜆_{2} = 2, 𝜆_{3} = 4