El-Sayed and Ran [17] studied the general matrix equation (X+ A^F X A=Q) where F maps positive definite matrices either into positive definite matrices or into negative definite matrices and satisfies some monotonicity property.
Note that K ~ 1 ( 1 ) is a valid choice of covariance matrix because it is the sum of positive semi-definite Hermitian matrices and satisfies tr { K ~ 1 ( 1 ) } = tr { K 1 ( 1 ) }. Since the determinant is a continuous function of the entries of the matrix, and the logarithm is a continuous function of its argument, we can find 0 < γ < 1 such that R ~ t ≜ α log I + H t H K ~ 1 ( 1 ) H t > R 1 ⋆.
We consider the nonnegative inverse eigenvalue problem with partial eigendata, which aims to find a nonnegative matrix such that it is nearest to a pre-estimated nonnegative matrix and satisfies the prescribed eigendata.
We next assume that n ≥ 2. If A is an M-matrix, then by Lemma 2.1 we know that there exists a diagonal matrix D with positive diagonal entries such that D − 1 A D is a strictly row diagonally dominant M-matrix and satisfies τ ( B ∘ A − 1 ) = τ ( D − 1 ( B ∘ A − 1 ) D ) = τ ( B ∘ ( D − 1 A D ) − 1 ).
(3.3) Based on Lemma 1, we know that for the vertex condition (( B_{1}mid cdotsmid B_{delta v)} ) hat{f}_{v}=0) of a self-adjoint operator (mathcal{L}), the coefficient matrix is a (delta v times2delta v)) matrix and satisfies the condition (B_{v}[S_{v}^{ast }]^{-1}B_{v}^{ast }=0 ).
We establish some criteria which guarantee that the above system has at least one or infinitely many homoclinic solutions under the assumption that (W t, x)) is subquadratic at infinity and (L t)) is a real symmetric matrix and satisfies liminf_{|t|to+infty} Bigl[|t|^{nu-2}inf _{|x|=1}bigl(L t)x, xbigr) Bigr] >0 for some constant (nu<2).
The actual control input implemented is assumed to be, where are known constant dimension matrices and satisfies.
Any matrix, where and, satisfies the assumptions in Theorem 2.1.
Two non-zero terms, M i,n (1) = t m,n and M i,m (1) = t n,m, exist at each row of matrix M i,j (1) and satisfy ∑ j M i,j (1) = 1.
