In the following, the Eqs. (12.4) and (12.5) will be transformed into the domain using the Laplace transform, which will be done analogously to the scalar case in section 2.5. For this, the operator notation

from section 2.1 is adopted and when applying it to Eq. (12.4), one obtains

or rearranged

The solution of the state equation in the domain is then given by

with

Similarly, for Eq. (12.5) yields

Substituting from Eq. (12.6), the system output in the domain is

To obtain the relationship with transfer functions, the initial condition has to be set to zero. For a single-input-single-output system according to Eqs. (12.2) and (12.3) the system output is

Comparing this equation with Eq. (3.3) the transfer function is given by

The matrix from Eq. (12.7) is a matrix of rational functions of , which can always be represented by

where is a matrix with polynomial elements in . From Eq. (3.2) it is obvious that

and

which is the characteristic polynomial of the system. The zeros of this polynomial are the poles of the transfer function and at the same time eigenvalues of the system matrix . If the system in the state-space representation is fully controllable and observable (see section 12.6), then the number of poles are equal to the number of eigenvalues.