Eigenvalues and eigenvectors

In linear algebra, various linear transformations may be applied to a vector, often affecting the magnitude and direction of the vector. However, a given transformation, $T$, can be associated with an eigenvector, a vector whose direction is unchanged by the transformation. While the direction remains unchanged, the eigenvector of a transformation is scaled by a corresponding eigenvalue, $\lambda$.

The linear transformation $A$ may have many eigenvectors. An eigenvector $\vec{v}$ of $A$ should satisfy the following:

$A\vec{v} = \lambda v$

Consider the matrix $\displaystyle A = \begin{bmatrix}2 & 1 \\ 1 & 2\end{bmatrix}$
Find the corresponding eigenvectors and eigenvalues for $A$.

• Linear algebra
• Vector