32.9. Summary of Key Concepts

Try these first

1. 1.

   Define an eigenspace in your own words.

   Hint

   Think about what null space means in context.

2. 2.

   What does it mean for a matrix to be diagonalizable?

   Hint

   Focus on how matrices can be expressed.

3. 3.

   What represents the null space of (A - λI)?

   • Eigenvectors
   • Eigenvalues
   • Eigenspaces

   Hint

   Consider the definition of eigenspaces.

4. 4.

   True or False: A matrix can be diagonalizable with fewer than n eigenvectors.

   • True
   • False

   Hint

   Reflect on the definition of diagonalizable matrices.

5. 5.

   Given the matrix A = [[2, 0], [0, 3]], find its eigenvalues and check if A is diagonalizable.

   Hint

   Calculate the eigenvalues and check the linear independence of the eigenvectors.

6. 6.

   Consider a 3x3 matrix with eigenvalues of 1, 1, and 2. Determine if it's diagonalizable.

   Hint

   Check the geometric vs algebraic multiplicity for eigenvalues.