📚 Topic Summary
Homogeneous systems of linear differential equations, in the form $\mathbf{x}' = A\mathbf{x}$, where $A$ is a constant matrix, can have solutions that involve repeated eigenvalues. When an eigenvalue $\lambda$ has an algebraic multiplicity greater than 1, but fewer linearly independent eigenvectors than its multiplicity, we must find generalized eigenvectors to form a complete set of solutions. These solutions often involve terms with $te^{\lambda t}$, $t^2e^{\lambda t}$, and so on, depending on the size of the repeated eigenvalue's multiplicity. Understanding how to find these generalized eigenvectors is key to solving these systems.
Let's put your knowledge to the test!
🧠 Part A: Vocabulary
Match the terms with their correct definitions:
| Term |
Definition |
| 1. Eigenvalue |
A. A non-zero vector that, when multiplied by a matrix, results in a scalar multiple of itself. |
| 2. Eigenvector |
B. A scalar value, $\lambda$, such that $A\mathbf{v} = \lambda\mathbf{v}$ for some non-zero vector $\mathbf{v}$. |
| 3. Algebraic Multiplicity |
C. A vector $\mathbf{v}$ satisfying $(A - \lambda I)^k \mathbf{v} = \mathbf{0}$ for some positive integer $k$, where $\lambda$ is an eigenvalue and $I$ is the identity matrix. |
| 4. Geometric Multiplicity |
D. The dimension of the eigenspace corresponding to a particular eigenvalue. |
| 5. Generalized Eigenvector |
E. The number of times an eigenvalue appears as a root of the characteristic equation. |
✍️ Part B: Fill in the Blanks
Fill in the blanks with the correct words:
When solving a homogeneous system with repeated eigenvalues, if the __________ multiplicity of an eigenvalue is greater than its __________ multiplicity, we need to find __________ eigenvectors. These vectors help us construct __________ independent solutions to the system.
🤔 Part C: Critical Thinking
Explain in your own words why finding generalized eigenvectors is necessary when dealing with repeated eigenvalues in a homogeneous system. What problem do they solve, and how do they contribute to the general solution?