Solved Examples: Finding Algebraic and Geometric Multiplicities

📚 Quick Study Guide

🔍 Algebraic Multiplicity: The algebraic multiplicity of an eigenvalue $\lambda$ is the number of times it appears as a root of the characteristic polynomial. To find it, calculate the characteristic polynomial $det(A - \lambda I)$ and factor it. 🔢 Characteristic Polynomial: Given a matrix $A$, the characteristic polynomial is defined as $p(\lambda) = det(A - \lambda I)$, where $I$ is the identity matrix. 📐 Geometric Multiplicity: The geometric multiplicity of an eigenvalue $\lambda$ is the dimension of the eigenspace associated with that eigenvalue. It is the number of linearly independent eigenvectors corresponding to $\lambda$. 💡 Finding Eigenspace: The eigenspace for an eigenvalue $\lambda$ is the null space of the matrix $(A - \lambda I)$. Solve the system $(A - \lambda I)v = 0$ to find the eigenvectors. 🧭 Relationship: $1 \le$ Geometric Multiplicity $\le$ Algebraic Multiplicity. This inequality always holds true. 📝 Diagonalizability: A matrix $A$ is diagonalizable if and only if the geometric multiplicity equals the algebraic multiplicity for all eigenvalues.

Practice Quiz

1. Question 1: What is the algebraic multiplicity of the eigenvalue $\lambda = 2$ for the matrix $A$ if its characteristic polynomial is $p(\lambda) = (\lambda - 2)^3(\lambda - 5)$?
   1. 1
   2. 2
   3. 3
   4. 4
2. Question 2: Which of the following statements is always true about the relationship between algebraic and geometric multiplicity?
   1. Algebraic multiplicity is always less than geometric multiplicity.
   2. Geometric multiplicity is always less than or equal to algebraic multiplicity.
   3. Algebraic multiplicity is always equal to geometric multiplicity.
   4. There is no relationship between algebraic and geometric multiplicity.
3. Question 3: For a matrix $A$, if an eigenvalue $\lambda$ has an algebraic multiplicity of 2 and a geometric multiplicity of 1, what can you conclude about the diagonalizability of $A$?
   1. $A$ is diagonalizable.
   2. $A$ is not diagonalizable.
   3. $A$ may or may not be diagonalizable; more information is needed.
   4. $A$ is invertible.
4. Question 4: If the eigenspace corresponding to an eigenvalue $\lambda$ is spanned by two linearly independent eigenvectors, what is the geometric multiplicity of $\lambda$?
   1. 1
   2. 2
   3. 3
   4. Cannot be determined.
5. Question 5: The characteristic polynomial of matrix $B$ is given as $(\lambda - 1)^2(\lambda - 3)$. The dimension of the eigenspace corresponding to $\lambda = 1$ is 1. What are the algebraic and geometric multiplicities of $\lambda = 1$, respectively?
   1. Algebraic: 1, Geometric: 2
   2. Algebraic: 2, Geometric: 1
   3. Algebraic: 1, Geometric: 1
   4. Algebraic: 2, Geometric: 2
6. Question 6: What is the first step in finding the algebraic multiplicity of an eigenvalue for a given matrix?
   1. Solve the system $(A - \lambda I)v = 0$.
   2. Find the determinant of $(A - \lambda I)$.
   3. Row reduce the matrix $A$.
   4. Find the trace of $A$.
7. Question 7: A 3x3 matrix $C$ has a single eigenvalue $\lambda = 4$ with algebraic multiplicity 3. If the geometric multiplicity of $\lambda = 4$ is also 3, is the matrix $C$ diagonalizable?
   1. Yes
   2. No
   3. Cannot be determined.
   4. Only if the matrix is symmetric.