Gramian determinant

Linear Spaces (Prentice-Hall, Englewood Cliffs, NJ, 1961), pp. 167-73] that the Gramian determinant... 

Gramian determinants and their ratios play a key role in the dimensional scaling treatment of many-electron atoms (and other many-body problems), so we summarize in this appendix several properties of these objects. 

Consider a set of N unit vectors f,- rooted at a common point, which can be taken to be the origin. Their Gramian determinant is defined as [27]... 

Now consider N vectors r,- of arbitrary lengths r,= rj. Although Gramian determinants are sometimes defined for non-unit vectors, it is more convenient for electronic structure applications to continue to keep the radial and angular aspects of the problem separated. Thus, we define the Gramian determinant for these vectors as above, with f = r,/r,-. The iV-dimensional volume of the parallelotope (t.e., generalized parallelepiped) defined by the r,- is clearly just rir2 ... 

The Gramian determinant ratios F d/r occurring in the centrifugal potential can also be expanded out in analogy with Eq. (34),... 

In the treatment of larger atoms, the evaluation of the Gramian determinants is the most time-consuming step. In the absence of assumptions regarding symmetry, it would undoubtedly be best to use elimination methods (requiring operations) rather than explicit expansions ( n operations) for the evaluations. However, when the matrices are highly structured, explicit expansions become more useful. This is the case, for example, if one assumes that all electrons within a shell are equivalent. Suppose that the atom has S shells, with N electrons in shell a. Then the Gramian determinant has the form... 

The Gramian determinant P is obtained by removing one row and one column from the matrix for F. All atoms in the ring (that is, the chain with periodic boimdary conditions) were equivalent, but it is simplest to eliminate the atom labeled 1 or N. This gives the Huckel determinant for a linear conjugated polyene [20], Cn-iH2n ... 

Determine the so-called controllability and observability gramians, the matrices P and Q. These are solutions to the equations ... 

If an adaptronic system is modelled in a state-space description (5.3), (5.4), its observability and controllability can be determined numerically by various methods. A common way is to compute the eigenvalues of the controllability and observability Gramians... 

High-order models are often a result of models consisting of many differential equations or partial differential equations that have been converted into ordinary differential equations. These types of model are adequate for simulations studies but are not suitable for online use. A popular technique of model reduction that does not make use of error minimization is the model balancing method. The procedure is to find observability and controllability Gramians so as to determine which states have the largest overall contribution to the model. In systems theory and linear algebra, a Gramian matrix is a real-values symmetric matrix that can be used for a test for linear independence of functions. A system is called controllable if all states X can be influenced by the control input vector w, a system is observable if all states can be determined from the measurement vector jp. 

Determinants of matrices of inner products as above are commonly called Gramians. 

Thus a four-point Cayley-Menger determinant is equal to the product of the signed volumes spanned by the parallelepipeds whose sides are the vectors from one of the points to the other three (in each of the two point sets separately). This is the syzygy that connects the squared distances to the signed volumes. The five-point Cayley-Menger determinants, on the other hand, are the Gramians of four interpoint vectors, and hence vanish identically whenever the distances are three-dimensional, as described at the beginning of this section. 

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