evaluated at matrix , and is the controllability matrix of the system. Proof This proof is based on Encyclopedia of Life Support Systems entry on Pole Placement Control. Assume that the system is controllable. The characteristic polynomial of is given by Calculating the powers of results in Replacing the previous equations into yieldsRewriting the above equation as a matrix product and omitting terms that does not appear isolated yields From the Cayley–Hamilton theorem, , thus Note that is the controllability matrix of the system. Since the system is controllable, is invertible. Thus, To find , both sides can be multiplied by the vector giving Thus, Example Consider We know from the characteristic polynomial of that the system is unstable since , the matrix will only have positive eigenvalues. Thus, to stabilize the system we shall put a feedback gain From Ackermann's formula, we can find a matrix that will change the system so that its characteristic equation will be equal to a desired polynomial. Suppose we want . Thus, and computing the controllability matrix yields and Also, we have that Finally, from Ackermann's formula State observer design Ackermann's formula can also be used for the design of state observers. Consider the linear discrete-time observed system with observer gain L. Then Ackermann's formula for the design of state observers is noted as with observability matrix . Here it is important to note, that the observability matrix and the system matrix are transposed: and . Ackermann's formula can also be applied on continuous-time observed systems. See also Full state feedback References External links Chapter about Ackermann's Formula on Wikibook of Control Systems and Control Engineering Mathematical concepts Engineering concepts Control engineering