Algebraic and Geometric Multiplicity Worksheets for University Linear Algebra

📚 Topic Summary

In linear algebra, eigenvalues and eigenvectors play a crucial role in understanding the behavior of linear transformations. Associated with each eigenvalue are two important concepts: algebraic multiplicity and geometric multiplicity. The algebraic multiplicity refers to the number of times an eigenvalue appears as a root of the characteristic polynomial. The geometric multiplicity, on the other hand, is the dimension of the eigenspace corresponding to that eigenvalue. Understanding the relationship between these two multiplicities is vital for analyzing the diagonalizability of a matrix and its behavior under repeated transformations.

This worksheet provides exercises to help you understand these concepts. Good luck!

🧮 Part A: Vocabulary

Match the terms with their definitions:

1. Term: Algebraic Multiplicity
2. Term: Geometric Multiplicity
3. Term: Eigenvalue
4. Term: Eigenspace
5. Term: Characteristic Polynomial

1. Definition: The set of all eigenvectors corresponding to a particular eigenvalue, together with the zero vector.
2. Definition: A scalar $\lambda$ such that $A\mathbf{v} = \lambda \mathbf{v}$ for some non-zero vector $\mathbf{v}$.
3. Definition: The dimension of the eigenspace associated with an eigenvalue.
4. Definition: The number of times an eigenvalue appears as a root of the polynomial $\det(A - \lambda I)$.
5. Definition: A polynomial whose roots are the eigenvalues of a matrix, given by $\det(A - \lambda I)$.

✍️ Part B: Fill in the Blanks

Complete the following sentences:

The algebraic multiplicity of an eigenvalue is always __________ than or equal to its geometric multiplicity. If the algebraic multiplicity equals the geometric multiplicity for all eigenvalues, then the matrix is __________ . The geometric multiplicity is found by calculating the __________ of $(A - \lambda I)$, where $\lambda$ is the eigenvalue and $I$ is the identity matrix. If an eigenvalue has an algebraic multiplicity of 3 and a geometric multiplicity of 1, the matrix is said to be __________ at that specific eigenvalue.

🤔 Part C: Critical Thinking

Explain, in your own words, why the geometric multiplicity of an eigenvalue can never be greater than its algebraic multiplicity. Provide an example to illustrate your reasoning.