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It is shown in this paper that any state space realization (A, b, c) of a given transfer function T(s)=β(s)/α(s) withα(s) monic and dim(A) = deg(α(s)), satisfies the identityβ(A) = Qc(A,b)SαQo(A,c) where Qc(A,b) and Qo(A,c) are the controllability matrix and observability matrix of the matrix triple (A,b,c), respectively, and Sαis a nonsingular symmetric matrix. Such an identity gives a deep relationship between the state space description and the transfer function description of single-input single-output (SISO) linear systems. As a direct conclusion, we arrive at the well-known result that a realization of any transfer function is minimal if and only if the numerator and the denominator of the transfer function is coprime. Such a result is also extended to the SISO descriptor linear system case. As an applications, a complete solution to the commuting matrix equation AX = XA is proposed and the minimal realization of multi-input multi-output (MIMO) linear system is considered.
It is shown in this paper that any state space realization (A, b, c) of a given transfer function T s = β s / α s with α sic and dim A = deg (a, c) where Qc (A, b) and Qo (A, c) are the controllability matrix and the observability matrix of the matrix triple (A, b, c), respectively, and Sαis a nonsingular symmetric matrix. Such an identity gives a deep relationship between the state space description and the transfer function description of single-input single-output (SISO) linear systems. As a direct conclusion, we arrive at the well-known result that a realization of any transfer function is minimal if and only if the numerator and the denominator of the transfer function is coprime. Such a result is also extended to the SISO descriptor linear system case. As an applications, a complete solution to the commuting matrix equation AX = XA is proposed and the minimal realization of multi-input multi-output (MIMO) linear system is considered.