Eigenvalues and eigenvectors - Eigenvalue and Eigenvectors

Notes:

An eigenvector of an $n \times n$ matrix $A$ is a non-zero 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 non-trivial 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 $A-\lambda I$.