Computing Eigenvectors

Eigenvectors are the unique, non-zero vectors that, when a linear transformation is applied to them, only scale in magnitude without changing direction. They represent the intrinsic directions along which a matrix acts purely as a scalar multiplier. Identifying these vectors is crucial in many applications, from principal component analysis to solving systems of differential equations, because they reveal the fundamental behavior of a linear transformation.

The process of finding eigenvectors begins with the eigenvalue equation, $Av = \lambda v$. This equation states that applying matrix $A$ to an eigenvector $v$ is equivalent to scaling $v$ by the eigenvalue $\lambda$. To solve for $v$, we rearrange this into a homogeneous system: $Av - \lambda v = 0$. Since $v$ is a vector, we cannot simply factor it out; instead, we introduce the identity matrix $I$ so that $\lambda v = \lambda I v$. This allows us to write the equation as $(A - \lambda I)v = 0$, which is a standard homogeneous linear system.

Before solving the system, you must first construct the matrix $(A - \lambda I)$. This involves subtracting the eigenvalue $\lambda$ from each element on the main diagonal of matrix $A$. All off-diagonal elements of $A$ remain unchanged. The identity matrix $I$ ensures that $\lambda$ is only applied to the diagonal elements, maintaining the correct matrix dimensions for subtraction.

Once you have the matrix $(A - \lambda I)$, the next step is to solve the homogeneous system $(A - \lambda I)v = 0$. This is equivalent to finding the null space of the matrix $(A - \lambda I)$. We typically use Gaussian elimination or row reduction to transform the matrix into its reduced row echelon form (RREF). The RREF will reveal the relationships between the components of $v$, often leading to free variables that allow for an infinite number of solutions, all of which are scalar multiples of a basis eigenvector.

When solving $(A - \lambda I)v = 0$, you often encounter free variables after row reduction. These variables can take any real value, leading to an infinite number of eigenvectors for a given eigenvalue. We express these solutions in terms of a parameter (e.g., $t$ or $s$), which defines the eigenspace — the subspace spanned by all eigenvectors corresponding to that $\lambda$. The basis vectors for this eigenspace are the specific eigenvectors we typically report.

Since any scalar multiple of an eigenvector is also an eigenvector, there are infinitely many eigenvectors for each eigenvalue. To ensure consistency and simplify comparisons, eigenvectors are often normalized to have a length (or magnitude) of 1. This process converts them into unit vectors. Normalization does not change the direction of the eigenvector, only its scale, making it a standard practice in many applications like Principal Component Analysis (PCA).

For each eigenvalue $\lambda$, the set of all corresponding eigenvectors (including the zero vector, which is technically not an eigenvector but is included for subspace properties) forms a vector subspace called the eigenspace $E_{\lambda}$. The dimension of this eigenspace is known as the geometric multiplicity of $\lambda$. When we compute eigenvectors, we are typically finding a basis for this eigenspace. If an eigenvalue has a geometric multiplicity greater than one (e.g., for repeated eigenvalues), its eigenspace will be spanned by multiple linearly independent eigenvectors.