33.10. Repeated Eigenvalues and Geometric Multiplicity

Try these first

1. 1.

   What is algebraic multiplicity?

   Hint

   Think about how many times you can factor the polynomial.

2. 2.

   If a matrix has a repeated eigenvalue, how can you check if it is diagonalizable?

   Hint

   Check the linear independence of the eigenvectors.

3. 3.

   What defines the algebraic multiplicity of an eigenvalue?

   • The number of linearly independent eigenvectors
   • The number of times an eigenvalue appears in the characteristic polynomial
   • The dimension of the eigenspace

   Hint

   Think about the factors of the polynomial.

4. 4.

   True or False: A matrix with repeated eigenvalues is always diagonalizable.

   • True
   • False

   Hint

   Consider an example to check this.

5. 5.

   Create a 3x3 matrix with one eigenvalue of multiplicity 3. Show its algebraic and geometric multiplicity and check if it's diagonalizable.

   Hint

   Start with the characteristic polynomial to find AM.

6. 6.

   Construct a matrix that is diagonalizable with two distinct eigenvalues, and verify its multiplicities.

   Hint

   Choose a simple diagonal matrix to illustrate diagonalizability.