29.3.2. Geometric Multiplicity

Try these first

1. 1.

   Define geometric multiplicity.

   Hint

   Think about the vectors corresponding to that eigenvalue.

2. 2.

   What is algebraic multiplicity?

   Hint

   Look at the characteristic polynomial.

3. 3.

   What defines geometric multiplicity?

   Hint

   Think about independent vectors corresponding to eigenvalues.

4. 4.

   True or False: The geometric multiplicity of an eigenvalue is always greater than its algebraic multiplicity.

   • True
   • False

   Hint

   Check the definitions of both multiplicities.

5. 5.

   A 3x3 matrix has an eigenvalue λ = 4 that appears twice in its characteristic polynomial. Determine the possible geometric multiplicities and discuss von implications on diagonalization.

   Hint

   Think about the definitions of multiplicities in regards to independence.

6. 6.

   Consider the matrix A = [[1, 1], [0, 1]]. Calculate the eigenvalues, their multiplicities and determine if A is diagonalizable.

   Hint

   Calculate the characteristic polynomial and confirm the case.