6. Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors

Definition and Characteristic Equation

Let $\mathbf{A}$ be an $n \times n$ matrix. A scalar $\lambda$ is called an eigenvalue of $\mathbf{A}$ if there exists a non-zero vector $\mathbf{v}$ such that:

$$ \mathbf{Av} = \lambda \mathbf{v} $$

The characteristic equation of $\mathbf{A}$ is given by:

$$ \text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0 $$

where $\mathbf{I}$ is the identity matrix.

Diagonalization and Eigenspaces

If an $n \times n$ matrix $\mathbf{A}$ has $n$ linearly independent eigenvectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$ corresponding to distinct eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$, then $\mathbf{A}$ can be diagonalized as:

$$ \mathbf{A} = \mathbf{PDP}^{-1} $$

where $\mathbf{P}$ is the matrix whose columns are the eigenvectors of $\mathbf{A}$ and $\mathbf{D}$ is the diagonal matrix whose diagonal elements are the eigenvalues of $\mathbf{A}$.

The eigenspace corresponding to an eigenvalue $\lambda$, denoted as $E_{\lambda}$, is the set of all eigenvectors corresponding to $\lambda$, along with the zero vector.

Applications to Systems of Differential Equations, Markov Chains, and Dynamical Systems

Eigenvalues and eigenvectors have various applications in different fields:

• Systems of Differential Equations: Eigenvalues and eigenvectors can be used to solve systems of linear ordinary differential equations by diagonalizing the coefficient matrix.
• Markov Chains: Eigenvalues and eigenvectors can be used to analyze the long-term behavior of Markov chains, where the transition matrix represents the probabilities of moving between states.
• Dynamical Systems: Eigenvalues and eigenvectors play a crucial role in analyzing the stability and behavior of linear dynamical systems described by differential equations.