Finding Eigenvectors for a Given Eigenvalue

Given this matrix $A$, find the eigenvector corresponding to the eigenvalue $\lambda = 3$.

Eigenvectors and eigenvalues are fundamental concepts in linear algebra that arise in a variety of applications, from stability analysis and vibrations to Google's PageRank algorithm. When dealing with matrices, an eigenvector of a matrix is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. This scalar factor is known as the eigenvalue associated with the eigenvector.

To solve for an eigenvector given an eigenvalue, one must first understand that the task involves solving a homogeneous system of linear equations, typically in the form of $(A - \lambda I) x = 0$, where $A$ is your matrix, $\lambda$ is the known eigenvalue, and $I$ is the identity matrix of the same size as $A$. This formulation converts the eigen-problem into a more familiar form: finding the null space of the matrix $(A - \lambda I)$. Conceptually, this means you are looking for vectors that remain in the same line during the transformation $A$, merely scaled by $\lambda$.