29.3. Algebraic and Geometric Multiplicities

Try these first

1. 1.

   Define algebraic multiplicity in your own words.

   Hint

   Think about the roots of a polynomial.

2. 2.

   What does geometric multiplicity represent?

   Hint

   Consider how many independent vectors relate to an eigenvalue.

3. 3.

   What does algebraic multiplicity refer to?

   • 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 how the characteristic polynomial is defined.

4. 4.

   Is the geometric multiplicity ever greater than the algebraic multiplicity?

   • True
   • False

   Hint

   Recall the inequality established in this section.

5. 5.

   Consider the 2x2 matrix A = | 1 2 | | 0 1 |. Determine the eigenvalues, their algebraic and geometric multiplicities, and discuss their implications for the matrix.

   Hint

   Examine the matrix closely using the determinant for λ.

6. 6.

   Create a 3x3 matrix with an eigenvalue having an algebraic multiplicity of 3 but a geometric multiplicity of 1. Analyze the eigenvalue's implications on linear transformations.

   Hint

   Focus on the structure of rows and columns leading to repeated eigenvalues.