32.14. Summary Table of Concepts

Try these first

1. 1.

   What is an eigenvector?

   Hint

   Remember that `λ` is the eigenvalue.

2. 2.

   Define eigenspace.

   Hint

   Think about the vectors that satisfy `Av = λv`.

3. 3.

   What defines an eigenvector?

   • A vector that changes direction under transformation.
   • A linear combination of vectors.
   • A non-zero vector satisfying Av = λv.

   Hint

   Focus on the eigenvalue equation.

4. 4.

   True or False: The eigenvalue's algebraic multiplicity can be greater than its geometric multiplicity.

   • True
   • False

   Hint

   Remember the definitions of both types of multiplicity.

5. 5.

   Given a matrix A = [[2, 1], [1, 2]], find its eigenvalues and eigenvectors. Discuss if it is diagonalizable.

   Hint

   Remember to use the determinant for the characteristic polynomial.

6. 6.

   Consider a symmetric matrix B with eigenvalues 4, 4, and 2. Determine the geometric and algebraic multiplicities, and explain the implications for diagonalization.

   Hint

   Think about how the multiplicities relate to the eigenspace dimensions.