Inner algorithm test for controllability and observability

In this note it is shown how the double triangularization algorithm developed for computing inners determinants can be adapted to test the controllability matrix (
B,AB,.., A^{n-1}B
) and observability matrix (
C^{T},A^{T}C^{T},...,(A^{T})^{n-1}C^{T}
) to have rank of "
n
." The controllability and observability conditions are shown to be equivalent to
n^{2} \times n(n + l - 1
) and
n(n + l' - 1) \times n^{2}
innerwise matrices to be nonsingular (i.e., to have a rank n
²
).