characteristic equation
The characteristic equation (C.E.) corresponding to the linear state space equations comprising the state equation?
x = Ax + Bu
and output equation
y = Cx + Du
is given by the determinant equation
|sI - A| = 0
This works out as a polynomial equation in s of order n when A is an n×n2 Accordingly it has n roots.Since A has real elements then the characteristic polynomial has real coeficients and so the roots occuras a set comprising real roots or complex conjugate pairs. For stability all roots must be in left half plane(ie all have negative real parts).