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On the matrix equation X+A T X -1 A=I. (English) Zbl 0863.15005
It is shown that the matrix equation (*) X+A T X -1 A=I has a positive definite solution (X>0) if and only if A has a factorization A=W T Z, where the matrix W is nonsingular and the columns of [W T ,Z T ] T are orthonormal. In this case X=W T W. It is also proved that equation (*) has a solution X>0 if and only if there exist orthogonal matrices P and Q and diagonal matrices Γ>0, Σ0 with Γ 2 +Σ 2 =I such that A=P T ΓQΣP. Finally, it is shown that if (*) has a solution X>0 then the following relations are valid X-AA T >0, I-AA T -A T A>0, r(A)1/2, r(A+A T )1, r(A-A T )1, where r(A) is the spectral radius of A.
MSC:
15A24Matrix equations and identities