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72 Chapter 3. Numeric solutions
Proof. Since S ≤ 0 and P, K ≥ 0, we have the following inequality: −A+SP ≤−A+SK,
and since −A + SK is a Z-matrix, Theorem 3.1.2 immediately gives, that −A + SK is also an M-matrix, which means that A − SK is stable. A sim- ilar argument shows also that −D + KS is an M-matrix too, hence
AT −K1S1 −K1S2 D−KS= −S1K2 AT −K2S2
is stable showing that K is left-right stabilizing. Remark 3.4.3. With the nonsingular matrices
In0 −1In0 T:= K I and T = −K I
2n 2n we obtain a spectral factorization of the Nash-matrix

T
−1 −1 A −S A−SK −S HNashT =T −Q −D T = 0 −D+KS

.
This reveals, together with the stability of A−SK and D−KS that HNash has exactly n eigenvalues in the open left half plane and 2n in the open right half plane. In particular, this implies that K is the dichotomic solution of the Nash algebraic Riccati equation (see [AFIJ03]). From the representation of solutions via HNash-invariant n-dimensional subspaces of R3n it is also immediate that K is the unique stabilizing solution.
Remark 3.4.4. Consider the Nash strategies:
ui =−R−1BiTKix∗, i=1,2,
where x∗ is given through the solution of the closed loop equation x˙ =(A−S1K1 −S2K2)x, x(0)=x0.
As the closed loop equation represents a positive system under the assumptions
of the main Theorem and above Corollary, the strategy functions ui are also
nonnegative, if R−1 ≤ 0. This condition is hidden in our assumption S ≤ 0 ii
since Bi ≥ 0 for i = 1, 2. In the considered framework the system remains in equilibrium also positive.
ii