6.2 Solving Systems using Eigenvalues and Eigenvectors

Systems of linear differential equations are solved using eigenvalues and eigenvectors. This method reveals how solutions behave over time, showing growth, decay, or oscillation. Understanding these concepts is key to analyzing complex systems in physics, engineering, and other fields.

The process involves finding eigenvalues, calculating eigenvectors, and using them to construct general solutions. This approach simplifies solving systems, providing insights into long-term behavior and stability. It's a powerful tool for tackling real-world problems in various disciplines.

Eigenvalues and Eigenvectors

Defining Eigenvalues and Eigenvectors

• Eigenvalues are scalar values $\lambda$ that satisfy the equation $A\vec{v} = \lambda\vec{v}$ for a square matrix $A$ and a non-zero vector $\vec{v}$
  • The values of $\lambda$ 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 the original vector ($A\vec{v} = \lambda\vec{v}$)
  • Eigenvectors maintain their direction when transformed by the matrix, only changing in magnitude by the corresponding eigenvalue

Calculating Eigenvalues and Eigenvectors

• The characteristic equation of a square matrix $A$ is given by $\det(A - \lambda I) = 0$, where $I$ is the identity matrix
  • Solving the characteristic equation yields the eigenvalues of the matrix
  • The degree of the characteristic polynomial equals the size of the square matrix
• Diagonalization is the process of decomposing a matrix $A$ into the product of three matrices: $A = PDP^{-1}$
  • $P$ is a matrix whose columns are the eigenvectors of $A$
  • $D$ is a diagonal matrix with the corresponding eigenvalues on its main diagonal
  • Diagonalization is possible if and only if the matrix $A$ has a full set of linearly independent eigenvectors

Types of Eigenvalues

Real Distinct Eigenvalues

• Real distinct eigenvalues occur when the characteristic equation has real roots that are all different from each other
  • For a 2x2 matrix, this means the discriminant of the characteristic equation is positive ($b^2 - 4ac > 0$)
• Systems with real distinct eigenvalues have exponential solutions that may grow, decay, or remain constant depending on the sign and magnitude of the eigenvalues
  • Example: A matrix with eigenvalues $\lambda_1 = 2$ and $\lambda_2 = -3$ will have solutions that grow exponentially in the direction of the first eigenvector and decay exponentially in the direction of the second eigenvector

Complex Conjugate Eigenvalues

• Complex conjugate eigenvalues occur when the characteristic equation has complex roots that appear in conjugate pairs ($a \pm bi$)
  • For a 2x2 matrix, this means the discriminant of the characteristic equation is negative ($b^2 - 4ac < 0$)
• Systems with complex conjugate eigenvalues have oscillatory solutions with exponential growth or decay depending on the real part of the eigenvalues
  • Example: A matrix with eigenvalues $\lambda_{1,2} = -1 \pm 2i$ will have solutions that oscillate with an exponentially decaying amplitude

Repeated Eigenvalues

• Repeated eigenvalues occur when the characteristic equation has roots with multiplicity greater than one
  • For a 2x2 matrix, this means the discriminant of the characteristic equation is zero ($b^2 - 4ac = 0$)
• Systems with repeated eigenvalues may have fewer linearly independent eigenvectors than the multiplicity of the eigenvalue
  • In this case, generalized eigenvectors must be found to complete the set of solutions
  • Example: A matrix with a repeated eigenvalue $\lambda = 3$ of multiplicity 2 may have only one linearly independent eigenvector, requiring the calculation of a generalized eigenvector to find the complete solution

Solving Systems using Eigenvalues

General Solution

• The general solution of a system of linear differential equations $\frac{d\vec{x}}{dt} = A\vec{x}$ is given by $\vec{x}(t) = c_1e^{\lambda_1t}\vec{v}_1 + c_2e^{\lambda_2t}\vec{v}_2 + \ldots + c_ne^{\lambda_nt}\vec{v}_n$
  • $\lambda_1, \lambda_2, \ldots, \lambda_n$ are the eigenvalues of the matrix $A$
  • $\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n$ are the corresponding eigenvectors
  • $c_1, c_2, \ldots, c_n$ are arbitrary constants determined by initial conditions
• The general solution represents a linear combination of exponential functions, each associated with an eigenvalue-eigenvector pair
  • The behavior of the solution depends on the types of eigenvalues (real distinct, complex conjugate, or repeated)

Fundamental Matrix

• The fundamental matrix $\Phi(t)$ is a matrix-valued function that satisfies the matrix differential equation $\frac{d\Phi}{dt} = A\Phi$ with the initial condition $\Phi(0) = I$
  • The columns of the fundamental matrix are linearly independent solutions to the system of differential equations
• If the matrix $A$ is diagonalizable, the fundamental matrix can be written as $\Phi(t) = Pe^{Dt}P^{-1}$
  • $P$ is the matrix of eigenvectors
  • $D$ is the diagonal matrix of eigenvalues
  • $e^{Dt}$ is a diagonal matrix with exponential functions of the eigenvalues on its main diagonal
• The solution to the system of linear differential equations with initial condition $\vec{x}(0) = \vec{x}_0$ can be written using the fundamental matrix as $\vec{x}(t) = \Phi(t)\vec{x}_0$
  • This representation simplifies the calculation of the solution by leveraging the properties of the fundamental matrix