31.1. Definition of Similar Matrices

Try these first

1. 1.

   Explain what it means for two matrices to be similar.

   Hint

   Think about how one matrix can represent the same transformation as another.

2. 2.

   Give an example of a property of similar matrices.

   Hint

   What happens when you consider P to be the identity matrix?

3. 3.

   What denotes that matrix A is similar to matrix B?

   • A = B
   • A ∼ B
   • A = P(B)
   • A * B = 0

   Hint

   Think about the mathematical notation used for similarity.

4. 4.

   If matrix A is similar to matrix B, is it true that matrix B is similar to matrix A?

   • True
   • False

   Hint

   Review the properties of similarity discussed.

5. 5.

   Prove that if matrices A and B are similar and B is a diagonal matrix, then A must also be diagonalizable.

   Hint

   Consider the implications of P's properties in relation to B's structure.

6. 6.

   Two matrices A and B are similar, and you know their characteristic polynomials. How would you show they have the same eigenvalues?

   Hint

   Refer to the definitions of eigenvalues in relation to characteristic polynomials.