Eigenvalues and Eigenvectors

We have been playing with systems of the form:

$$A\vec{x}=\vec{b}$$

Now what if:

$$\vec{b}=\lambda \vec{x}$$

Basically, $\vec{b}$ is a scaler multiple of a non-zero $\vec{x}$.

If such a scaler $\lambda$ exists for a non-zero $\vec{x}$, it is called the eigenvalue of $A$. And $\vec{x}$ is the corresponding eigenvector of $A$.

• We want $\vec{x}$ to be a non-zero vector as $A\vec{0}=\lambda \vec{0}$ would be true for any scalar $\lambda$.