MATH319 Slides 57 Realization with a SISO 59 Cofactors

58 Proof of realization

We require to prove $T(s)=C(sI-A)^{-1}B$ . Recall that

(sI-A)^{-1}=\det(sI-A)^{-1}{\hbox{adj}}(sI-A)

where the adjugate is the transpose of the matrix of cofactors. Also ${\hbox{adj}}(sI-A)B$ equals the last column of ${\hbox{adj}}(sI-A)$ , so by transposition, ${\hbox{adj}}(sI-A)B$ equals the final row of the matrix of cofactors of $sI-A$ , where

sI-A=\begin{bmatrix}s&-1&0&\dots&0\\ 0&s&-1&\ddots&0\\ \vdots&\ddots&\ddots&\ddots&\vdots\\ \vdots&\ddots&\ddots&s&-1\\ \alpha_{0}&\alpha_{1}&\dots&\alpha_{n-2}&s+\alpha_{n-1}\end{bmatrix}

We compute these one after another. Recall that the determinant of an upper or lower triangular matrix equals the product of the diagonal entries.