Sentence examples for minor determinants from inspiring English sources

The principal minor determinants are used to determine if a matrix is positive or negative definite or semidefinite.

To determine if H is positive or negative definite or semidefinite, its principal minor determinants are derived as D 0 = 2 x i y i (34) D 1 = 2 x i y i x i y j + x j y i x i y j + x j y i 2 x j y j = - ( x i y j - x j y i ) 2 (35) D 2 = D 3 = ⋯ = D n = 0, where 1 ≤ i < j ≤ n. (36).

Changes in forest landscape composition due to afforestation and fires were minor determinants of range changes, and forest management did not seem to prevent range expansion at the spatial scale studied.

Lemma 2.2 Suppose all the principal minor determinants of A are non-negative.

Lemma 2.3 Suppose that all the lower-order principal minor determinants of A are non-negative and A is irreducible.

Lemma 2.4 Suppose that all the lower-order principal minor determinants of A are non-negative and |A| < 0.

In addition, pillar dimensions serve as a minor determinant of the observed differences in nuclear shape.

It is possible to see that the Hessian matrix of the objective, the first-order principal minor determinant, and the second-order principal minor determinant, have the same structure as the 2-CTP case.

Therefore, the Hessian ∇2Γ is indefinite since both, the first-order and the second-order principal minor determinant are negative.

The first-order principal minor determinant and the second-order principal minor determinant have the same structure as the 2-CTP case, which means the Hessian ∇2Γ is indefinite since both the first and second-order principal minor determinant are negative.

The first principal minor determinant of H is begin{aligned} |H_{11}|=frac{4D}{Q^{*3}}left[ frac{(S+eta rho ^m+nF)}{left{ 2-theta (a+b right} }right] quad quad quad quad ({rm A}1) end{aligned}The second principal minor of H is begin{aligned} |H_{22}|&=frac{4D}{Q^{*3}}left[ frac{(S+eta rho ^m+nF)}{left{ 2-theta (a+b right} }right] frac{2tau M}{S^2}+frac{2D}{ Q^{*2} 2-theta (a+b))}nonumber &>0.