# Eigenvalues and eigenvectors of a matrix.

I needed to know what are the eigenvalues and eigenvectors of this matrix:

$$\left[\begin{array}{cc}+\cos\theta&-\sin\theta\\ +\sin\theta&+\cos\theta\end{array}\right]$$

First time having this matter, so I'm sorry if this sounds too easy

• Have you tried using the usual method for finding eigenvalues? Jan 14 '14 at 10:43
• This is a $2\times 2$ rotation matrix.. Eigenvalues/eigenvectors are easy to search for. Jan 14 '14 at 10:43
• take the determinant of $(A-I\lambda)$, solve the characteristic polynomial for lambda, and go from there Jan 14 '14 at 10:44

The eigenvalues $\lambda$ of the matrix can be easily found by solving:

$$|M-\lambda I| = 0,$$

where $I$ is the $2\times 2$ unit matrix. This equations give us the spectrum of $M$:

$$\lambda_{1,2} = \cos{\theta} \pm i \sin{\theta}$$

On the other hand, the eigenvectors are to be determined as the nullspace of the following matrices:

$$\vec{v}_{1,2} = \text{Ker}(M-\lambda_{1,2}I) = \left\{ (x,y)\in \mathbb{C}/ \ (M-\lambda_{1,2}) \left(\begin{array}{cc} x \\ y \end{array} \right) = 0 \right \},$$

$$\vec{v}_1 = (i,1), \quad \vec{v}_2 = (i,-1)$$