The eigenvalues are calculated by substituting (PDB-1, PDB-3, PDB-5 and UHB-LP) values of the system equilibrium points into values of state variables in the Jacobian matrix
In this study, for the non-linear transformation from rectangular coordinate to polar coordinate, Jacobian matrix
The basic Jacobian matrix
is used to express the kinematics torsor [?
Newton's method requires the Jacobian matrix
to be computed at each iteration.
In that respect, one of the key advantages of CV over even AD is its extreme simplicity and practicality for all scales computations of the gradient, Jacobian matrix
, as well as the Hessian and Laplacian.
F(x), a n x n Jacobian matrix
of partial derivative whenever [x.
If 'm' number of satellites are observed at a given time, Jacobian matrix
J will be mx4 matrix, R will be mx1 matrix and [delta]x would be 4x1 matrix.
For systems, we have to give a combinatorial meaning to the inverse of the Jacobian matrix
m], usually requires the repeated calculation or estimation of the matrix of first derivatives, the Jacobian matrix
, J(x) [element of] [R.
The various measures of interaction strength can be placed into four theoretical categories: elements in the community matrix (Levins 1968, MacArthur 1972), elements in the Jacobian matrix
(May 1973), elements in the inverse Jacobian matrix
(Levine 1976, Yodzis 1988), and elements in the "removal" matrix (following MacArthur 1972 and Paine 1992).
Based on literature , Jacobian matrix
of the differential equation system made of (4) and (5) is
where f(x) is Jacobian matrix
of n-dimensional function at point [x.