Symmetric matrix, properties

Another useful property of the S matrix is that it is symmetric. This property follows from conservation of the fluxlike expression... 

Ihe metric matrix G is a square symmetric matrix. A general property of such matrices that they can be decomposed as follows ... 

The matrices [G] and [F] are column matrices with row numbers n and k, respectively. The matrix solution is simplified by special properties of the symmetric matrix and because the resulting values of G occur in complex conjugate pairs. In general, we may write... 

A square matrix is one in which the number of columns is equal to the number of rows. An important type of square matrix which arises quite often in the finite element method is a symmetric matrix. Such matrices possess the property that aij = aji- An example of such a matrix is given below ... 

Proof.—There exists a unitary skew symmetric matrix C with the property that... 

Statistical properties of a data set can be preserved only if the statistical distribution of the data is assumed. PCA assumes the multivariate data are described by a Gaussian distribution, and then PCA is calculated considering only the second moment of the probability distribution of the data (covariance matrix). Indeed, for normally distributed data the covariance matrix (XTX) completely describes the data, once they are zero-centered. From a geometric point of view, any covariance matrix, since it is a symmetric matrix, is associated with a hyper-ellipsoid in N dimensional space. PCA corresponds to a coordinate rotation from the natural sensor space axis to a novel axis basis formed by the principal... 

Each of these columns of this symmetrical matrix may be seen as representing a molecule in the subspace formed by the density functions of the N molecules that constitute the set. Such a vector may also be seen as a molecular descriptor, where the infinite dimensionality of the electron density has been reduced to just N scalars that are real and positive definite. Furthermore, once chosen a certain operator in the MQSM, the descriptor is unbiased. A different way of looking at Z is to consider it as an iV-dimensional representation of the operator within a set of density functions. Every molecule then corresponds to a point in this /V-dimensional space. For the collection of all points, one can construct the so-called point clouds, which allow one to graphically represent the similarity between molecules and to investigate possible relations between molecules and their properties [23-28]. 

An(p) must be a symmetric matrix in the spaces of internal degrees of freedom. Taking into account the property that the most attractive channel of the OGE interaction is the color antisymmetric 3 state, it must be in the flavor singlet state. 

Here nig denotes the sign of the spin projection (it takes two values, +1 and — 1). By taking into account the cumulant properties, Eqs. (10)-(13) with replaced by F, it can be shown that A must be a real symmetric matrix (A " = A j) with no unique diagonal elements, whereas 11 is a spin-independent (11 = =... 

If there are no other IRs common to Y and X, then eqs. (15) show that there are only four non-zero Krs coefficients in K. Depending on the nature of the property it represents, K may be a symmetric matrix, and if this is true here K 2 = X2U leaving three independent coefficients. Equations (13)—(15) hold also for E or T representations since according to the FT the K coefficients that belong to different degeneracy indices are zero, while the coefficients for the same d are independent of d. 

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. 

The Lanczos method is based on generating the orthonormal basis in Krylov space Ki =span c, Ac, A c by applying the Gram-Schmidt orthogonaliza-tion process, described in Appendix A. In matrix notations this approach is associated with the reduction of the symmetric matrix A to a tridiagonal matrix and also with the special properties of T/,. This reduction (called also QT decomposition) is described by the formula... 

A symmetric matrix has two important properties (1) all of its characteristic roots are real and (2) it has n independent orthogonal characteristic directions (vectors) 72). Thus, the desired conclusions follow immediately when the proof is obtained that the reaction system can be transformed into a similar system with a symmetrical rate constant matrix. This proof also provides the tools for proving the characteristic roots are nonpositive (negative or zero). Furthermore, it also provides the transformation to an orthogonal characteristic system of coordinates useful for consistency checks and for obtaining the inverse of the matrix X. 

If all N N-1)/2 distances in a figure are given, it is possible to find a set of Cartesian coordinates for the N vertices. The method makes use of the properties of the metric matrix. First, the connectivity matrix C, the symmetric NxN matrix of distances squared is formed. From it we form an (Af-l)x(Af-l) symmetric matrix G by subtracting the first row and column from all other rows and columns, respectively. The diagonal elements are then of the form -2(cfi ) and the off-diagonal ones are which is the same as from the properties of the... 

Properties of the Real Symmetric Matrix H = ArA (A is Real, Nonsingular, and of Order n)... 

The orthonormality properties of the two functions can be represented by the symmetric matrix... 

Interestingly the time-reversal symmetry provides analogous properties in some respects to the properties of Hamiltonian systems. For example, consider a linear Hamiltonian system dz/dl = JAz, with A a symmetric matrix. If A is an eigenvalue of JA then JAu = Xu for some eigenvector u 0. Because the matrix JA is real, we know that A will also be an eigenvalue. At the same time, we know that since A is an eigenvalue of JA it is also an eigenvalue of its transpose (JAY = A J = —AJ, thus... 

There is a separate solution corresponding to each possible incoming channel, and the solution is characterized at long range by the S-matrix with elements Sji. The S-matrix is an A open x A open complex symmetric matrix, where A open is the number of open channels. It is unitary, that is, SS = I, where indicates the Hermitian conjugate and / is a unit matrix. If the physical problem is factorized into separate sets of coupled equations for different symmetries (such as total angular momentum or parity), there is a separate S-matrix for each symmetry. All properties that correspond to completed collisions, such as elastic and inelastic integral and differential cross-sections, can be written in terms of S-matrices. 

---