ODE: Repeated Eigenvalues and Generalized Eigenvectors

When solving systems of linear ordinary differential equations, you typically rely on the eigenvalue-eigenvector method to find fundamental solutions. But what happens when your coefficient matrix is defective, meaning it doesn't have a full set of linearly independent eigenvectors? This scenario, common in engineering models for vibrations, control systems, and circuit analysis, forces you to move beyond standard methods. To find all solutions, you must understand the relationship between algebraic and geometric multiplicity, master the computation of generalized eigenvectors, and construct solutions that involve polynomial-exponential terms, a direct consequence of the underlying Jordan normal form.

Algebraic Multiplicity, Geometric Multiplicity, and Defective Matrices

The core challenge begins with the eigenvalues. For a given eigenvalue $\lambda$, its algebraic multiplicity is the number of times it appears as a root of the characteristic polynomial $p(\lambda )=\det (A-\lambda I)$. In contrast, its geometric multiplicity is the dimension of the corresponding eigenspace, which is the number of linearly independent eigenvectors you can find for that $\lambda$. This number is given by $\text{dim}\text{Nul}(A-\lambda I)$.

A matrix is considered defective when for at least one eigenvalue, the geometric multiplicity is strictly less than the algebraic multiplicity. For a system of ODEs ${\mathbf{x}}^{\prime }=A\mathbf{x}$, this defect means you cannot form a complete basis of solutions from just the "pure" exponential solutions $e^{\lambda t}\mathbf{v}$. The system is missing solutions, and you must find them. The gap between the algebraic multiplicity ($m$) and the geometric multiplicity ($k$) tells you precisely how many missing solutions you need to generate for that eigenvalue: you will need to find $m-k$ generalized eigenvectors.

Computing Generalized Eigenvectors and Building Jordan Chains

A generalized eigenvector of rank $r$ associated with eigenvalue $\lambda$ is a nonzero vector ${\mathbf{v}}_r$ that satisfies $(A-\lambda I{)}^r{\mathbf{v}}_r=0$, but $(A-\lambda I{)}^{r-1}{\mathbf{v}}_r\ne 0$. The most important ones for solving ODEs are those of rank 2, which fill the most common gaps.

You find them by constructing a Jordan chain. Start with a genuine eigenvector ${\mathbf{v}}_1$ (a rank-1 generalized eigenvector). To find a rank-2 generalized eigenvector ${\mathbf{v}}_2$, you solve the equation: $$(A-\lambda I){\mathbf{v}}_2={\mathbf{v}}_1.$$ This equation is key. It often has infinitely many solutions; you typically choose the simplest one. If the gap is larger, you continue the chain: $(A-\lambda I){\mathbf{v}}_3={\mathbf{v}}_2$, and so on. Each vector in the chain is linearly independent. The length of the chain for a given eigenvector equals the size of the Jordan block associated with that eigenvalue in the Jordan normal form.

Constructing the General Solution: Polynomial-Exponential Terms

The power of generalized eigenvectors is that they generate the missing, linearly independent solutions to your ODE system. For a chain of length 2 starting with an eigenvector ${\mathbf{v}}_1$ and a generalized eigenvector ${\mathbf{v}}_2$, you generate two fundamental solutions:

1. The standard solution from the eigenvector: $e^{\lambda t}{\mathbf{v}}_1$.
2. The new solution generated by the generalized eigenvector: $e^{\lambda t}(t{\mathbf{v}}_1+{\mathbf{v}}_2)$.

Notice the polynomial term ($t$) multiplying the exponential. If you had a longer chain of length 3 (${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$), the corresponding solutions would be: $$e^{\lambda t}{\mathbf{v}}_1,e^{\lambda t}(t{\mathbf{v}}_1+{\mathbf{v}}_2),e^{\lambda t}\left(\frac{t^2}{2}{\mathbf{v}}_1+t{\mathbf{v}}_2+{\mathbf{v}}_3\right).$$

The pattern is systematic: each successive generalized eigenvector in the chain introduces a higher polynomial degree in $t$, divided by the appropriate factorial. The general solution to ${\mathbf{x}}^{\prime }=A\mathbf{x}$ is a linear combination of all such solutions from all eigenvalues and their associated chains.

Connection to Jordan Normal Form and the Fundamental Matrix

This entire procedure is the practical implementation of Jordan normal form theory. The Jordan form $J$ of a matrix $A$ is obtained via the similarity transformation $A=PJ{P}^{-1}$, where $P$ is the modal matrix whose columns are precisely the eigenvectors and generalized eigenvectors you computed, arranged in their chains.

For solving the ODE, this decomposition is incredibly powerful. The substitution $\mathbf{x}=P\mathbf{z}$ transforms the system into ${\mathbf{z}}^{\prime }=J\mathbf{z}$. Because $J$ is block-diagonal (with Jordan blocks), this new system is partially decoupled and easily solved, leading directly to the polynomial-exponential solutions. The fundamental matrix $X(t)=e^{At}$ can be computed as $X(t)=P{e}^{Jt}{P}^{-1}$, where $e^{Jt}$ has a clear, block-wise structure. This theoretical backbone guarantees that the method of generalized eigenvectors will always yield a complete set of solutions for a constant-coefficient linear system.

Common Pitfalls

1. Stopping at Eigenvectors: The most critical error is assuming that because you found an eigenvalue of algebraic multiplicity $m$, you must find $m$ independent eigenvectors. You must check the dimension of the eigenspace. If $\text{dim}\text{Nul}(A-\lambda I)<m$, you must begin searching for generalized eigenvectors.
2. Incorrect Chain Order: When solving $(A-\lambda I){\mathbf{v}}_2={\mathbf{v}}_1$, it is essential that the right-hand side is a true eigenvector. Starting from a random vector that is not in the null space of $(A-\lambda I)$ will not produce a valid chain. Always verify that your ${\mathbf{v}}_1$ satisfies $(A-\lambda I){\mathbf{v}}_1=0$.
3. Misconstructing the Solution: A common computational mistake is to incorrectly assemble the polynomial terms. Remember the pattern: the solution derived from generalized eigenvector ${\mathbf{v}}_k$ in a chain includes terms with $t^{k-1},t^{k-2},...$ down to $t^0$, each multiplying the appropriate chain vector. For a rank-2 vector, the solution is specifically $e^{\lambda t}(t{\mathbf{v}}_1+{\mathbf{v}}_2)$, not $e^{\lambda t}(t{\mathbf{v}}_2)$.
4. Overlooking Simpler Solutions: When solving for a generalized eigenvector, the linear system $(A-\lambda I){\mathbf{v}}_2={\mathbf{v}}_1$ is always consistent but underdetermined. You have freedom in choosing a particular solution. Often, you can set a free variable to zero to obtain a vector with simple, often integer, components to make subsequent calculations cleaner.

Summary

• The gap between an eigenvalue's algebraic multiplicity (its root count) and its geometric multiplicity (the dimension of its eigenspace) diagnoses a defective matrix and signals the need for generalized eigenvectors.
• Generalized eigenvectors are found by solving a sequence of equations starting from a true eigenvector: $(A-\lambda I){\mathbf{v}}_k={\mathbf{v}}_{k-1}$. This builds a Jordan chain.
• Each generalized eigenvector in a chain generates an independent solution to ${\mathbf{x}}^{\prime }=A\mathbf{x}$ of the form $e^{\lambda t}$ times a polynomial in $t$ whose coefficients are the vectors in the chain.
• The complete set of eigenvectors and generalized eigenvectors forms the matrix $P$ that puts $A$ into its Jordan normal form $J$, which fully explains the structure of the system's solutions.
• The final general solution is a linear combination of standard exponential solutions from eigenvectors and polynomial-exponential solutions from generalized eigenvectors.