Eigenvalues and eigenvectors  Eigenvalue and Eigenvectors
Eigenvalues and eigenvectors
Lessons
Notes:
An eigenvector of an $n \times n$ matrix $A$ is a nonzero vector $x$ such that $Ax= \lambda x$, for some scalar $\lambda$. The scalar $\lambda$ is called the eigenvalue.
We say the eigenvector $x$ corresponds to the eigenvalue $\lambda$.
Given an eigenvalue $\lambda$ of matrix $A$, we can find a corresponding eigenvector $x$ by solving
$(A\lambda I)x=0$
And finding a nontrivial solution $x$.
The eigenspace is the null space of the matrix $A\lambda I$. In other words, the eigenspace is a set of all solutions for the equation
$(A\lambda I)x=0$
Of course, we can find the basis for the eigenspace by finding the basis of the null space of $A\lambda I$.

Intro Lesson
Eigenvalues and Eigenvectors Overview: