29.5. Diagonalization of a Matrix

Try these first

1. 1.

   What does it mean for a matrix to be diagonalizable?

   Hint

   Consider what forms the P and D matrices take.

2. 2.

   Define eigenvector in your own words.

   Hint

   Think about how vectors behave under matrix multiplication.

3. 3.

   What is a diagonalizable matrix?

   • A matrix with all zero eigenvalues
   • A matrix that can be expressed as A = PDP⁻¹
   • A matrix that is only square

   Hint

   Think about the definition provided earlier.

4. 4.

   True or False: The diagonal matrix D only contains elements on its diagonal.

   • True
   • False

   Hint

   Remember the definition of a diagonal matrix.

5. 5.

   Given a matrix A = [[4, 1], [1, 4]], find if it is diagonalizable, and if so, compute the diagonalized form.

   Hint

   Start by finding the characteristic polynomial to determine eigenvalues.

6. 6.

   If a system of differential equations can be expressed in matrix form as dx/dt = Ax, how would diagonalization of A assist in solving this system?

   Hint

   Consider how each eigenvalue influences the shape of the solution curve.