M. Fiedler and V. Ptak. Some generalizations of positive definiteness and monotonicity. Numerische Mathematik, 9:163--172, 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....#(k 1 , km ) k i # # i , # i , 41) all have the same nonzero sign. When the intervals [# i , # i ] are replaced by (0, #) the condition (40) is the definition of D being a P 0 matrix , and there is a similar necessary and su#cient condition for D to be a P 0 matrix [FP62, FP66]. It is clear that this condition is necessary, since the image of = # 1 , # 1 ] # m , #m ] 42) under # is connected, and therefore an interval, so if it contains numbers of di#erent signs, it contains zero. To prove su#ciency we will show that the maximum and minimum of # over are ....

M. Fiedler and V. Ptak. Some generalizations of positive definiteness and monotonicity. Numerische Mathematik, 9:163--172, 1966.

....For every nonzero vector x 2 R n , there exists an index i such that x i (Mx) i 0. 5 (iii) The implication x 2 R n ; x (Mx) 0 ) x = 0 (1) holds, where x (Mx) is the componentwise product of vectors x and Mx. iv) For every q 2 R n , LCP(M; q) has a unique solution. We recall from [5] that a matrix M 2 R n Thetan is a P matrix (or is said to have the P property) if it satisfies condition (i) or equivalently, either condition (ii) or condition (iii) Thus, in the LCP setting, the uniqueness of solution in LCP(M; q) is described by the P property of the matrix M . In this ....

M. Fiedler and V. Pt'ak, Some generalizations of positive definiteness and monotonicity, Numer. Math. 9: 163-172 (1966).

....is together with a W 0 property significant to investigate properties mentioned above. Keywords M matrices, interval matrices, P matrices 0.1 Introduction A matrix M 2 R n Thetan is called a P matrix if every principal minor of M is positive. This notion, introduced by Fiedler and Pt ak [FP66], have numerous applications in diverse fields. The class of P matrices is not closed under product of matrices. To study this problem we use an extension of Wilson s concept [Wil71] which introduced a notion of W 0 pair of matrices to study certain equations arising in nonlinear DCnetworks. We ....

Miroslav Fiedler and Vlastimil Pt'ak. Some generalizations of positive definiteness and monotonicity. Numerische Mathematik, 9:163--172, oct 1966.

....in matrix form as below. For a vector x, the notation x 0 will mean that all coordinates of x are strictly positive, and the notation x 0 will mean that all coordinates of x are non negative. Definition 1.3 A matrix A is called an S matrix if there is a vector x 0 such that Ax 0. See [18] for more details on S matrices which are named after Stiemke. Remark. It is easy to see by perturbation that in the definition of an S matrix, x may be chosen such that x 0. Let N denote the m Theta d matrix whose i th row is given by the row vector n 0 i for each i 2 J. For an m ....

Fiedler, M. and Pt'ak, V. Some generalizations of positive definiteness and monotonicity, Numerische Mathematik 9, 163--172 (1966).

....prove. 2 Remark. In the Nonlinear Complementarity Problem NCP (G) H(z) Gammaz) the hypothesis of Theorem 4.1 reads: For all u 2 IR q ; u 6= 0, there exists a diagonal matrix D(u ) with strictly positive diagonal entries such that u T D(u )G 0 (y ) T u 0. Friedler and Pt ak [13] proved that this hypothesis holds if G 0 (y ) is a P Gammamatrix (all principal minors of G 0 (y ) are positive) Let us recall (see [4] that a matrix A 2 IR n Thetan is called an S matrix if there exists x 2 IR n ; x 0 such that Ax 0. It can be proved (see [13] that A is an ....

....Friedler and Pt ak [13] proved that this hypothesis holds if G 0 (y ) is a P Gammamatrix (all principal minors of G 0 (y ) are positive) Let us recall (see [4] that a matrix A 2 IR n Thetan is called an S matrix if there exists x 2 IR n ; x 0 such that Ax 0. It can be proved (see [13]) that A is an S matrix if, and only if, fy 2 IR n j y 0; y 6= 0; A T y 0g is empty. A matrix A 2 R n Thetan is said to be column sufficient (see [4] if for all u 2 R n such that [u] i [Au] i 0 for all i = 1; n one necessarily has that [u] i [Au] i = 0 for all i = 1; ....

M. Friedler and V. Pt'ak [1966], Some generalizations of positive definiteness and monotonicity. Numerische Mathematik 9, pp. 163-172.

....) taken from C(A; B) that is, M = A; B) S 0 [ EnS) 00 ) and N = A; B) EnS) 0 [ S 00 ) we have that each real root of jM Gamma N j is non negative. 6. There exists a complementary pair (M; N ) taken from C(A; B) such that M Gamma1 N 2 P 0 , in the sense of Fiedler and Pt ak [14]. 7. There exists a non singular M 2 C(A; B) and for any complementary pair (M;N ) taken from C(A; B) with M non singular, M Gamma1 N 2 P 0 . A pair of matrices (A; B) that satisfies these properties is called a W 0 pair, denoted (A; B) 2 W 0 . This section shows two conditions on the ....

M. Fiedler and V. Pt'ak, Some generalizations of positive definiteness and monotonicity, Numer. Math., 9, 163-172, (1966).

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