Engineering Analysis/Cayley Hamilton Theorem

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 than S, then the matrix A itself is also a solution to that function:

$f(A)=\sum _{i=0}^{\infty }b_{i}A^{i}$