Eigen-Stuff

Definitions

Eigenvalue and Eigenvector

Let $T: V \rightarrow V$ be a linear transformation on a vector space $V$ over a field $\mathbb{F}$. A scalar $\lambda \in \mathbb{F}$ is called an eigenvalue of $T$ if there exists a non-zero vector $v \in V$ such that:

The non-zero vector $v$ is called an eigenvector corresponding to the eigenvalue $\lambda$.

Eigenspace

For a given eigenvalue $\lambda$ of $T$, the eigenspace $E_\lambda$ is the set of all vectors that satisfy $T(v) = \lambda v$, including the zero vector:

The eigenspace is a subspace of $V$.

Properties

Finding Eigenvalues and Eigenvectors

The eigenvalues of a matrix $A$ are found by solving the characteristic polynomial:

For each eigenvalue $\lambda$, the corresponding eigenvectors are the non-zero solutions to:

Multiplicity

• The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial.
• The geometric multiplicity of an eigenvalue is the dimension of its eigenspace.

Important Theorems

1. The sum of the algebraic multiplicities of all eigenvalues equals the size of the matrix.
2. The geometric multiplicity of an eigenvalue is always less than or equal to its algebraic multiplicity.
3. A matrix is diagonalizable if and only if the geometric multiplicity equals the algebraic multiplicity for every eigenvalue.

Examples

2x2 Matrix

Consider the matrix:

1. Find eigenvalues:

2. Find eigenvectors:
   • For $\lambda = 1$: $(A - I)v = 0 \Rightarrow v = \begin{pmatrix} 1 \ -1 \end{pmatrix}$
   • For $\lambda = 3$: $(A - 3I)v = 0 \Rightarrow v = \begin{pmatrix} 1 \ 1 \end{pmatrix}$

Applications

Eigenvalues and eigenvectors are fundamental in:

• Solving systems of differential equations
• Principal Component Analysis (PCA) in statistics
• Stability analysis in physics and engineering
• Quantum mechanics (energy levels and states)
• Graph theory (spectral graph theory)
• Computer graphics (principal axes of objects)