3.2 Eigenvalues and eigenvectors

Eigenvalues and eigenvectors are crucial in understanding linear systems. They help us analyze how matrices transform vectors and provide insights into system behavior. These concepts are key to solving differential equations and studying stability.

In this section, we'll learn how to calculate eigenvalues and eigenvectors, explore their properties, and see how they're used in real-world applications. We'll also look at special cases like complex eigenvalues and repeated eigenvalues.

Eigenvalues and Eigenvectors

Defining Eigenvalues and Eigenvectors

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• Eigenvalues are scalar values $\lambda$ associated with a linear system of equations $A\vec{v} = \lambda\vec{v}$
  • $A$ is a square matrix and $\vec{v}$ is a non-zero vector
  • Eigenvalues represent the scaling factor by which the eigenvector is transformed when multiplied by the matrix
• Eigenvectors are non-zero vectors $\vec{v}$ that, when multiplied by a square matrix $A$, result in a scalar multiple of themselves $A\vec{v} = \lambda\vec{v}$
  • Eigenvectors maintain their direction when transformed by the matrix, only changing in magnitude
  • For a given eigenvalue, the corresponding eigenvector is not unique and can be scaled by any non-zero constant

Calculating Eigenvalues and Eigenvectors

• The characteristic equation $\det(A - \lambda I) = 0$ is used to find the eigenvalues of a square matrix $A$
  • $I$ is the identity matrix of the same size as $A$
  • Expanding the determinant leads to a polynomial equation in $\lambda$, known as the characteristic polynomial
  • The roots of the characteristic polynomial are the eigenvalues of the matrix
• To find the eigenvectors corresponding to an eigenvalue $\lambda$, solve the equation $(A - \lambda I)\vec{v} = \vec{0}$
  • This equation represents a homogeneous system of linear equations
  • Non-trivial solutions to this system are the eigenvectors associated with the eigenvalue $\lambda$

Properties and Applications

• The sum of the eigenvalues of a matrix equals the trace (sum of the diagonal elements) of the matrix
• The product of the eigenvalues equals the determinant of the matrix
• Eigenvalues and eigenvectors have numerous applications in physics, engineering, and computer science
  • Vibration analysis (natural frequencies and modes of a system)
  • Stability analysis of dynamical systems (stable, unstable, or neutral equilibria)
  • Principal component analysis (data compression and feature extraction)
  • Quantum mechanics (energy levels and stationary states of a system)

Complex Eigenvalues

• Eigenvalues of a real matrix can be complex numbers
  • Complex eigenvalues always occur in conjugate pairs (if $a + bi$ is an eigenvalue, then $a - bi$ is also an eigenvalue)
• Eigenvectors corresponding to complex eigenvalues are also complex
  • Real and imaginary parts of the eigenvectors separately satisfy the eigenvector equation
• Systems with complex eigenvalues exhibit oscillatory behavior
  • The real part determines the growth or decay of the oscillation
  • The imaginary part determines the frequency of the oscillation

Diagonalization and Special Cases

Diagonalization

• A square matrix $A$ is diagonalizable if it can be written as $A = PDP^{-1}$
  • $D$ is a diagonal matrix containing the eigenvalues of $A$
  • $P$ is a matrix whose columns are the corresponding eigenvectors of $A$
  • $P^{-1}$ is the inverse of $P$
• Diagonalization simplifies matrix operations and analysis
  • Powers of a diagonalizable matrix can be easily computed: $A^n = PD^nP^{-1}$
  • Exponential of a diagonalizable matrix: $e^A = Pe^DP^{-1}$, where $e^D$ is a diagonal matrix with $e^{\lambda_i}$ on the diagonal
• A matrix is diagonalizable if and only if it has a full set of linearly independent eigenvectors

Repeated Eigenvalues

• A repeated eigenvalue (or multiple eigenvalue) is an eigenvalue with algebraic multiplicity greater than one
  • Algebraic multiplicity is the number of times the eigenvalue appears as a root of the characteristic polynomial
• The geometric multiplicity of an eigenvalue is the dimension of its corresponding eigenspace (number of linearly independent eigenvectors)
  • Geometric multiplicity is always less than or equal to the algebraic multiplicity
• A matrix with repeated eigenvalues is diagonalizable if and only if the geometric multiplicity equals the algebraic multiplicity for each eigenvalue

Generalized Eigenvectors

• When the geometric multiplicity is less than the algebraic multiplicity, generalized eigenvectors are used to complete the basis
• A generalized eigenvector $\vec{v}$ satisfies the equation $(A - \lambda I)^k\vec{v} = \vec{0}$ for some positive integer $k$
  • $k$ is the smallest positive integer for which this equation holds
  • Generalized eigenvectors are not eigenvectors in the usual sense, as they do not satisfy the standard eigenvector equation
• Generalized eigenvectors, along with the eigenvectors, form a basis for the matrix and can be used in the Jordan canonical form

Advanced Topics

Jordan Canonical Form

• The Jordan canonical form (JCF) is a matrix decomposition that extends the concept of diagonalization to matrices that are not diagonalizable
• A matrix $A$ can be written in its Jordan canonical form as $A = PJP^{-1}$
  • $J$ is a block diagonal matrix called the Jordan matrix
  • Each block in $J$ is a Jordan block associated with an eigenvalue
• A Jordan block $J_i(\lambda)$ is a square matrix of the form: $J_i(\lambda) = \begin{pmatrix} \lambda & 1 & 0 & \cdots & 0 \\ 0 & \lambda & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \\ 0 & 0 & 0 & \cdots & \lambda \end{pmatrix}$
• The size of each Jordan block is determined by the geometric multiplicity of the corresponding eigenvalue
• The matrix $P$ in the Jordan decomposition consists of the eigenvectors and generalized eigenvectors of $A$
• The Jordan canonical form simplifies the computation of matrix functions and the analysis of systems with repeated eigenvalues
  • Powers of a matrix in JCF: $A^n = PJ^nP^{-1}$, where $J^n$ is obtained by raising each Jordan block to the power $n$
  • Exponential of a matrix in JCF: $e^A = Pe^JP^{-1}$, where $e^J$ is obtained by exponentiating each Jordan block