Systems of Linear ODEs with Eigenvalues Cheat Sheet

Key Facts

• A homogeneous linear system is written as $\mathbf{x}' = A\mathbf{x}$, where $\mathbf{x}(t)$ is a vector of unknown functions and $A$ is a constant matrix.
• Eigenvalues are found from the characteristic equation $\det(A - \lambda I) = 0$.
• For a real eigenvalue $\lambda$ with eigenvector $\mathbf{v}$, one solution is $\mathbf{x}(t) = e^{\lambda t}\mathbf{v}$.
• If $A$ has $n$ linearly independent eigenvectors $\mathbf{v}_1,\ldots,\mathbf{v}_n$, then $\mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{v}_1 + \cdots + c_n e^{\lambda_n t}\mathbf{v}_n$.
• For a repeated eigenvalue $\lambda$ with one eigenvector $\mathbf{v}$ and generalized eigenvector $\mathbf{w}$ satisfying $(A - \lambda I)\mathbf{w} = \mathbf{v}$, two solutions are $e^{\lambda t}\mathbf{v}$ and $e^{\lambda t}(t\mathbf{v} + \mathbf{w})$.
• For complex eigenvalues $\lambda = \alpha \pm i\beta$ with eigenvector $\mathbf{v} = \mathbf{p} + i\mathbf{q}$, real solutions come from $e^{\alpha t}\left(\mathbf{p}\cos(\beta t) - \mathbf{q}\sin(\beta t)\right)$ and $e^{\alpha t}\left(\mathbf{p}\sin(\beta t) + \mathbf{q}\cos(\beta t)\right)$.
• The sign of $\operatorname{Re}(\lambda)$ determines growth or decay: $\operatorname{Re}(\lambda) < 0$ gives decay, $\operatorname{Re}(\lambda) > 0$ gives growth, and $\operatorname{Re}(\lambda) = 0$ gives neutral oscillation in the linear model.
• Initial conditions are applied by substituting $t = 0$ into the general solution and solving for the constants $c_1,c_2,\ldots,c_n$.

Vocabulary

Linear system

A set of differential equations that can be written in matrix form as $\mathbf{x}' = A\mathbf{x}$ for a constant matrix $A$.

Eigenvalue

A scalar $\lambda$ such that $A\mathbf{v} = \lambda\mathbf{v}$ for some nonzero vector $\mathbf{v}$.

Eigenvector

A nonzero vector $\mathbf{v}$ that keeps its direction under multiplication by $A$, changing only by the factor $\lambda$.

Generalized eigenvector

A vector $\mathbf{w}$ used for defective repeated eigenvalues, satisfying $(A - \lambda I)\mathbf{w} = \mathbf{v}$.

Fundamental solution set

A linearly independent collection of solution vectors that can be combined to form every solution of the system.

Phase portrait

A geometric picture of solution curves in the state plane showing how the vector $\mathbf{x}(t)$ evolves over time.