32.4. Algebraic and Geometric Multiplicity

Try these first

1. 1.

   Define Algebraic Multiplicity.

   Hint

   Relate it to counting occurrences.

2. 2.

   What does Geometric Multiplicity represent?

   Hint

   Think about the space spanned by eigenvectors.

3. 3.

   What is algebraic multiplicity?

   • Number of linearly independent eigenvectors
   • Dimension of eigenspace
   • Times eigenvalue appears in the characteristic polynomial

   Hint

   Think about counting citations in a research paper.

4. 4.

   True or False: The geometric multiplicity can be greater than the algebraic multiplicity.

   • True
   • False

   Hint

   Reflect on their definitions.

5. 5.

   For a matrix A with eigenvalues and their respective multiplicities: λ1=2, AM=3, GM=2; λ2=0, AM=1, GM=1. Discuss the diagonalizability of A.

   Hint

   Refer back to the conditions that define diagonalizability and the role of eigenvalue multiplicities.

6. 6.

   Explain why a symmetric matrix has only real eigenvalues. How does this impact AM and GM?

   Hint

   Consider how the properties of symmetrical matrices simplify eigenvalue calculations.