29.3.1. Algebraic Multiplicity

Try these first

1. 1.

   What is algebraic multiplicity?

   Hint

   Consider how many times you can factor the polynomial.

2. 2.

   Define geometric multiplicity.

   Hint

   Think about how many directions correspond to that eigenvalue.

3. 3.

   What does algebraic multiplicity refer to?

   • The number of eigenvectors for an eigenvalue
   • The number of times an eigenvalue is a root of the characteristic polynomial
   • The stability of the matrix

   Hint

   Focus on root counting!

4. 4.

   True or False: Geometric multiplicity can be greater than algebraic multiplicity.

   • True
   • False

   Hint

   Remember the inequalities we've discussed.

5. 5.

   Consider a matrix with characteristic polynomial p(λ) = (λ - 3)³(λ - 2) = 0. Determine the algebraic and geometric multiplicities for each eigenvalue.

   Hint

   Don't forget the concepts and check eigenspace dimensions!

6. 6.

   If an eigenvalue's algebraic multiplicity is 5 and geometric is 3, what can you infer about the matrix properties in terms of stability?

   Hint

   Link it back to applications in structural assessments.