Let T=[A B;C D] be a real (m+n)X(m+n) stable matrix with controllable pair (D,C). Choosing such matrices T uniformly at random:
What is the probability that the nonsymmetric algebraic Riccati equation
XCX-XD+AX-B=0 has solutions ?
Coveying a lot of computer experiments, I always came up with a solution to the above mensioned nonsymmetric algebraic Riccati equation.
On the other hand, for A=-I, D=-I,C=I, B=[0 1;0 0], the matrix T is stable (with eigenvalue equals to -1) and (D,C) is controllable, but there is no solution for
X^{2}=B.
I conjecture that the measure of the set of such matrices among all matrices T as above is 0, but I cant find any way to proove it.
Can anyone suggest a good refference to such questions ?
Dear Yossi,
first, you may omit the condition that T=[A B; C D] is a STABLE (block) matrix. In fact, the matrix equation XCX-XD+AX-B=0 is equivalent to the equation
[I X; 0 0].T.[0 X; 0 -I]=0. Since [I X; 0 0].cI.[0 X; 0 -I]=0 (direct checking) for any real constant c then [I X; 0 0].(T+cI).[0 X; 0 -I]=0.
Second, (just an approach, not a solution). Let's fix a given matrix X and consider the equation as a linear (homogeneous) equation on matrix entries of T. Thus we obtain a mapping S: X->S(X) where S(X) is the subspace of solutions in (m+n)^2-dimensional real space of matrices T. The problem is to describe (in terms of measure) the image of the mapping S.
Best regards, Yuri
I think, if you can compute the eigenspace of Hamiltonian matrix associated to this equation. If you need some reference of that i can send it to you.
Dear Yuri and dear Agoujil !
Many thanks for you help !
To Agoujil: the algebraic Riccati here is nonsymetric and the "Hamiltonian" matrix here is T itself. Is there any theorem of eigenvalues separation related to the nonsymmetric case? If so, could you please help me with this ?
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