# Engineering Analysis/Cayley Hamilton Theorem

If the characteristic equation of matrix A is given by:

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

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

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

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:

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

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:

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