33.2. Eigenvalues and Eigenvectors Review

Try these first

1. 1.

   What is the definition of an eigenvalue?

   Hint

   Think of the transformation of vectors.

2. 2.

   How do you find eigenvalues for a matrix?

   Hint

   Calculate the determinant and set it to zero.

3. 3.

   What does an eigenvalue represent?

   • A vector that changes direction
   • A scalar that indicates scaling of an eigenvector
   • A constant value of the matrix

   Hint

   Think about what happens when a vector is transformed.

4. 4.

   The equation Av = λv is used to define which concept?

   • True
   • False

   Hint

   Recall the definition of eigenvalues and eigenvectors.

5. 5.

   Given the matrix B = [[5, 4], [0, 2]], find its eigenvalues and eigenvectors, and discuss its diagonalizability.

   Hint

   Start with the characteristic equation.

6. 6.

   Consider the matrix C = [[1, 1], [0, 1]]. Investigate if it's diagonalizable and explain why or why not.

   Hint

   Check the multiplicities!