31.7. Examples

Try these first

1. 1.

   What does it mean for two matrices to be similar?

   Hint

   Think about the idea of changing coordinate systems.

2. 2.

   How do you find eigenvalues from a matrix?

   Hint

   Remember the characteristic equation involves the determinant!

3. 3.

   What condition must be satisfied for two matrices to be considered similar?

   • They must have the same determinant
   • They must have the same eigenvalues
   • They must be orthogonal

   Hint

   Recall the criteria for matrix similarity.

4. 4.

   True or False: A matrix can be similar to a matrix that is not diagonalizable.

   • True
   • False

   Hint

   Think about the diagonalizability condition for similarity.

5. 5.

   Determine if the matrix D = [[1, -2], [0, 1]] is diagonalizable. Justify your answer with calculations.

   Hint

   Check the eigenvectors found and see if any are lost due to multiplicity.

6. 6.

   For matrices F and G, if F is similar to G and G = [[3, 0], [0, 1]], what can you infer about the eigenvalues of F?

   Hint

   Recall how similar matrices share eigenvalues.