Advanced examples: Solving (A - λI)x = 0 for repeated eigenvalues.

📚 Quick Study Guide

• 🔢 When solving $(A - \lambda I)x = 0$ for repeated eigenvalues, the algebraic multiplicity (the number of times the eigenvalue is a root of the characteristic polynomial) may be greater than the geometric multiplicity (the dimension of the eigenspace).
• 🔑 If the algebraic multiplicity equals the geometric multiplicity for all eigenvalues, the matrix is diagonalizable.
• 📝 If the algebraic multiplicity is greater than the geometric multiplicity for some eigenvalues, the matrix is not diagonalizable, and you need to find generalized eigenvectors.
• ➗ To find generalized eigenvectors, solve $(A - \lambda I)^k x = 0$ for $k = 2, 3,...$ until you find enough linearly independent eigenvectors and generalized eigenvectors to span the entire vector space.
• 💡 The vectors obtained will form a chain, and you can use them to construct the matrix $P$ for the transformation.

🧪 Practice Quiz

1. What does it mean if the algebraic multiplicity of an eigenvalue is greater than its geometric multiplicity?
   1. A) The matrix is diagonalizable.
   2. B) The matrix is not diagonalizable.
   3. C) The eigenvalue is not real.
   4. D) The determinant of the matrix is zero.
2. What equation do you solve to find generalized eigenvectors?
   1. A) $(A - \lambda I)x = 0$
   2. B) $(A - \lambda I)^2 x = 0$
   3. C) $(A - \lambda I)^k x = 0$ for $k = 2, 3,...$
   4. D) $Ax = 0$
3. If you have a $3 \times 3$ matrix with an eigenvalue of algebraic multiplicity 3 and geometric multiplicity 1, how many linearly independent eigenvectors and generalized eigenvectors do you need to find?
   1. A) 1
   2. B) 2
   3. C) 3
   4. D) 4
4. What is the purpose of finding generalized eigenvectors?
   1. A) To diagonalize the matrix.
   2. B) To find a basis for the eigenspace.
   3. C) To find a set of linearly independent vectors that span the entire vector space when the matrix is not diagonalizable.
   4. D) To calculate the determinant of the matrix.
5. Suppose $(A - \lambda I)v_1 = 0$ and $(A - \lambda I)v_2 = v_1$. What is the relationship between $v_1$ and $v_2$?
   1. A) $v_1$ is an eigenvector, and $v_2$ is a generalized eigenvector.
   2. B) $v_2$ is an eigenvector, and $v_1$ is a generalized eigenvector.
   3. C) Both $v_1$ and $v_2$ are eigenvectors.
   4. D) Neither $v_1$ nor $v_2$ are eigenvectors.
6. If the geometric multiplicity of an eigenvalue equals the algebraic multiplicity, what can you conclude about the matrix?
   1. A) The matrix is invertible.
   2. B) The matrix is diagonalizable.
   3. C) The matrix has complex eigenvalues.
   4. D) The matrix is not diagonalizable.
7. What is the first step in solving $(A - \lambda I)x = 0$ for repeated eigenvalues?
   1. A) Find the determinant of $A$.
   2. B) Find the eigenvalues of $A$.
   3. C) Find the trace of $A$.
   4. D) Find the inverse of $A$.