31. Similarity of Matrices

Try these first

1. 1.

   What does it mean for two matrices to be similar?

   Hint

   Think about the conditions for similarity.

2. 2.

   State one property of similar matrices.

   Hint

   Review the invariant properties.

3. 3.

   What is matrix similarity?

   • A relation between two non-square matrices
   • A relation where two matrices represent the same linear transformation
   • A relation involving only diagonal matrices

   Hint

   Focus on the definition of connection between different bases.

4. 4.

   True or False: If two matrices are similar, they must be diagonalizable.

   • True
   • False

   Hint

   Consider the definitions and implications of diagonalizability.

5. 5.

   Given matrix A = [[1, 2], [0, 1]] and matrix B = [[1, 1], [0, 1]], verify if A and B are similar through finding the appropriate matrix P.

   Hint

   Consider the effect of each matrix on a vector in their respective bases.

6. 6.

   Construct a 3x3 matrix that is not diagonalizable but can still demonstrate similarity to a diagonal matrix.

   Hint

   Think about the Jordan form when constructing this matrix.