Matrix algebra positivity

The representation of an EMB(A) by its fee-matrix B = (b ), an n x n symmetric matrix with positive integer entries and well-defined algebraic properties, corresponds to the act of translating an object of chemistry into a genuine mathematical object. This implies that the chemistry of an EMB(A) corresponds to the algebraic properties of the matrix B. 

When ordering must be carried out automatically, two very widely used techniques for symmetric positive definite matrices are CuthiU-McKee Ordering and Reverse CuthiU-McKee Ordering. The Cuthill-McKee method is usually described by means of the graph theory (Saad, 2003). It is also possible to realize it by means of matrix algebra, as illustrated by the simple example below. 

A careful reader will observe that this algebraic transformation will produce a dual SDP problem that does not have y G IR such that the matrix in Eq. (16) has all of its eigenvalues positive and, therefore, will not satisfy the Slater conditions. However, numerical experiments have shown that practical algorithms stUl can solve these problems efhciently [16]. 

Making use of the properties of the eigenvalues of Casimir operators, mentioned in Chapter 5, we are in a position to find a number of interesting features of the matrix elements of the Coulomb interaction operator. Thus, it has turned out that for the pN shell there exists an extremely simple algebraic expression for this matrix element... 

The set xfis is referred to as the basis set, the basis functions being normally chosen to be centred at the nuclei and to depend only on the positive charge of the nucleus. The chosen functions may be atomic orbitals of the component atoms, but are not necessarily so. Variation of the total energy with respect to the coefficients c,n leads to a set of algebraic equations which can be written in matrix form ... 

Symmetry operations may be represented by algebraic equations. The position of a point (an atom of a molecule) in a Cartesian coordinate system is described by the vector r with the components x, y, z. A symmetry operation produces a new vector r with the components t, /, z. The algebraic expression representing a symmetry operation is a matrix. A symmetry operation is represented by matrix multiplication. 

Since all differences in Eq. (4.73) are positive according to Fig. 4.20,3 this equation is satisfied under all circumstances. Therefore, it is unconditionally stable. We obtain this stability, however, at the cost of a new algebraic complexity. Recall Eq. (4.50), in which all temperatures except T[ +l are known, and the latter is obtained by solution of the equation. By contrast, in Eq. (4.72) only T is known, and the application of this equation to the nodes yields a tridiagonal matrix which then can be solved by the method discussed in Ex. 4.1. 

Again notice the sign convention that products have positive coefficients and reactants have negative coefficients in Equations 2.6. Equations 2.6 now resemble a set of three linear algebraic equations and motivates the use of matrices. Using the rules of matrix multiplication, one can express Equations 2,6 as... 

Higham, N. J., Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Applic., 103, 103 (1988). 

If S S is to be negative for all possible variations 8li and 5V, then the matrix S must be negative definite or equivalently, (-S) must be positive definite. The conditions under which S is negative definite are given by a theorem from linear algebra it is necessary and sufficient that the principal minors of S satisfy the following inequalities [5] ... 

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