A multiple-input multiple-output system is written in terms of the component input vector

(1)

generating a set of simultaneous linear equations, where the matrix is . The component system output vector may be found by substituting this solution for into the output equation as in Eq. 2:

(2)

and expanding the inverse in terms of the determinant and the adjoint matrix

(3)

where is defined to be the matrix transfer function relating the output vector to the input vector :

(4)

For a system with inputs and outputs is a matrix whose elements are individual scalar transfer functions relating a given component of the output to a component of the input . Expansion of Eq. 3 generates a set of equations:

(5)

where the th component of the output vector is:

(6)

The elemental transfer function is the scalar transfer function between the thoutput component and the th input component. Equation 5 shows that all of the transfer functions in have the same denominator factor , giving the important result that all input-output differential equations for a system have the same characteristic polynomial, or alternatively have the same coefficients on the left-hand side.

If the system has a single-input and a single-output, is a scalar, and the procedure generates the input/output transfer operator directly.