32. Basis of Eigenvectors

Try these first

1. 1.

   Define an eigenvector.

   Hint

   Think about the relationship between **A**, **v**, and **λ**.

2. 2.

   What is the formula for determining the eigenspace associated with an eigenvalue?

   Hint

   Consider the equation form involving the matrix **A**.

3. 3.

   What defines an eigenvector?

   • A vector that can only be zero
   • A vector that satisfies Av = λv
   • Any random vector

   Hint

   Focus on the defining equation involving the matrix **A**.

4. 4.

   True or False: The eigenspace of an eigenvalue consists solely of the eigenvector itself.

   • True
   • False

   Hint

   Consider whether the eigenspace could have multiple vectors.

5. 5.

   Consider the matrix B = [[5, 4], [2, 3]]. Compute its eigenvalues and eigenvectors and discuss their significance in terms of modulating the behavior of a system.

   Hint

   Set up the characteristic equation and solve for λ.

6. 6.

   A matrix has three eigenvalues: 2, 2, and 3. What can you say about its diagonalizability based on geometric and algebraic multiplicities?

   Hint

   Realize the importance of the corresponding eigenspaces.