29.12. Cayley-Hamilton Theorem

Try these first

1. 1.

   What is the Cayley-Hamilton theorem?

   Hint

   It relates to eigenvalues.

2. 2.

   What does the characteristic polynomial represent?

   Hint

   Consider how determinants are calculated.

3. 3.

   What does the Cayley-Hamilton theorem state?

   • A matrix cannot satisfy equations.
   • Every square matrix satisfies its own characteristic equation.
   • Only symmetric matrices satisfy it.

   Hint

   It relates to how eigenvalues operate in matrices.

4. 4.

   True or False: The characteristic polynomial is always a second-degree polynomial.

   • True
   • False

   Hint

   Consider the number of dimensions in square matrices.

5. 5.

   Consider a 2x2 matrix A with eigenvalues λ_1 = 3 and λ_2 = 5. Derive the characteristic polynomial and apply the Cayley-Hamilton theorem to find A^2.

   Hint

   Consider what happens with the polynomial’s roots when you evaluate using the matrix A.

6. 6.

   Given a matrix B, describe how to apply the Cayley-Hamilton theorem to find the inverse of B efficiently when B is large.

   Hint

   How does the theorem help reduce the complexity of matrix inversion?