Engineering Analysis/Cayley Hamilton Theorem

      Engineering Analysis

      If the characteristic equation of matrix A is given by:

      \Delta(\lambda) = |A-\lambda I| = (-1)^n(\lambda^n + a_{n-1}\lambda^{n-1} + \cdots + a_0) = 0

      Then the Cayley-Hamilton theorem states that the matrix A itself is also a valid solution to that equation:

      \Delta(A) = (-1)^n(A^n + a_{n-1}A^{n-1} + \cdots + a_0) = 0

      Another theorem worth mentioning here (and by "worth mentioning", we really mean "fundamental for some later topics") is stated as:

      If λ are the eigenvalues of matrix A, and if there is a function f that is defined as a linear combination of powers of λ:

      f(\lambda) = \sum_{i = 0}^\infty b_i \lambda^i

      If this function has a radius of convergence S, and if all the eigenvectors of A have magnitudes less then S, then the matrix A itself is also a solution to that function:

      f(A) = \sum_{i = 0}^\infty b_i A^i
      Last modified on 17 April 2007, at 19:03