# Multiplication of matrices

Multiplication of matrices The produet of two matriees [c.427]

The matrix equation [E.l] involves the multiplication of the matrices [A] and (x). To do this one must apply the simple rules of matrix multiplication. These are [c.432]

Thus, the matrices will have a multiplication table with the same structure as the multiplication table of the synnnetry group and hence will fonn an /-dimensional representation of the group. [c.158]

Clusters are intennediates bridging the properties of the atoms and the bulk. They can be viewed as novel molecules, but different from ordinary molecules, in that they can have various compositions and multiple shapes. Bare clusters are usually quite reactive and unstable against aggregation and have to be studied in vacuum or inert matrices. Interest in clusters comes from a wide range of fields. Clusters are used as models to investigate surface and bulk properties [2]. Since most catalysts are dispersed metal particles [3], isolated clusters provide ideal systems to understand catalytic mechanisms. The versatility of their shapes and compositions make clusters novel molecular systems to extend our concept of chemical bonding, stmcture and dynamics. Stable clusters or passivated clusters can be used as building blocks for new materials or new electronic devices [4] and this aspect has now led to a whole new direction of research into nanoparticles and quantum dots (see chapter C2.17). As the size of electronic devices approaches ever smaller dimensions [5], the new chemical and physical properties of clusters will be relevant to the future of the electronics industry. [c.2388]

For efficiency the number of Gaussian functions used must be kept as small as possible, otherwise time spent building and inverting the matrices will become prohibitive. The big question is where to put the Gaussian functions for the initially unoccupied state to ensure that they are present in regions of strong non-adiabatic coupling when required. The multiple spawning method does this by generating new functions in non-adiabatic regions when required, that is, when the wavepacket enters the region [218]. [c.296]

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In computed tomography (CT) (98) the usual x-ray film image is replaced by sets of digitized matrices which represent the x-ray attenuation through the body. Multiple x-ray projections are utilized. After the data are computer-analyzed, cross-sectional views of the target organ(s) can be generated. The advantage of CT over the more conventional x-ray imaging technique is the greater contrast sensitivity to attenuation changes. However, because film is a continuous medium whereas the CT images are derived from digital picture elements (pixels), resolution of very small stmctures generated from a finite number of pixels can be limited usiag CT, as compared to coaveatioaal film-screen radiography. [c.469]

Partial least-squares path modeling with latent variables (PLS), a newer, general method of handling regression problems, is finding wide apphcation in chemometrics. This method allows the relations between many blocks of data ie, data matrices, to be characterized (32—36). Linear and multiple regression techniques can be considered special cases of the PLS method. [c.426]

Multiplication Let A = ( 7 ), i = 1,. . . , mi J= I,. . . , mo. B = (by), i= I,. . . , ni,j = I,. . . , Ho. The produc t AB is defined if and only if the number of columns of A (mo) equals the number of rows of B ni), i.e., til = mo. For two such matrices the product P = AB is defined by summing the element by element products of a row of A by a column of B. This is the row by column rule. Thus [c.465]

The emerging model for protein dynamics is one that incorporates the dual aspects of motion within minima, combined with transitions between minima. The normal mode analysis can be seen as addressing directly only one of these two features. Ironically, however, one variant of nonnal mode analysis can be used to help address the other feamre, namely the transition of energy barriers. To determine barrier heights for transitions occurring in a molecular dynamics simulation, instantaneous normal modes can be determined by diagonalizing instantaneous force matrices at selected configurations along the trajectory. Negative eigenvalues indicate local negative curvature possibly arising from energy barriers in the multiple-minima surface. Simulations performed at different temperatures can give information on the distribution of barrier heights [59,60]. [c.163]

Once a multiple alignment is constructed, matrices of pairwise sequence similarities are usually calculated and employed to construct a phylogenetic tree that expresses the relationships among the proteins in the family [60]. All significantly different structures in the cluster that contains the target sequence are usually used as templates in the subsequent model building [61], although other considerations should also enter into the template selection. For example, if the model is prepared to study the liganded state of a protein, then a template in the liganded state is preferred over a template without a ligand. Some methods allow short segments of known structure, such as loops [62], to be added to the alignment at this stage [31]. [c.280]

The skills matrices shown as Figures 2-8 and 2-9 are provided as an example of one way to approach this task. The skills indicated are examples only there may be others your company s initiative might need. The first matrix. Figure 2-8, helps to locate skills within your company. Note that as a practical matter, one person may have multiple skills, just as several people may share one key skill. Note also that this matrix can help identify "pockets" or concentrations [c.35]

Multiplication of two matrices (AB) is only possible if the number of columns in A is equal to the number of rows in B. If A is an m x n matrix and B is an n x o matrix then the product AB is an m X 0 matrix. Each element (i, ) in the matrix AB is obtained by taking each of the n values in the ith row of A and multiplying by the corresponding value in the )th colunrn of B. To illustrate with a simple example [c.33]

The conceptually simplest approach to solve for the -matrix elements is to require the wavefimction to have the fonn of equation (B3.4.4). supplemented by a bound function which vanishes in the asymptote [32, 33, 34 and 35] This approach is analogous to the fiill configuration-mteraction (Cl) expansion in electronic structure calculations, except that now one is expanding the nuclear wavefimction. While successfiti for intennediate size problems, the resulting matrices are not very sparse because of the use of multiple coordinate systems, so that this type of method is prohibitively expensive for diatom-diatom reactions at high energies. [c.2295]

A general treatment of quantum reaction dynamics for multiple interacting elechonic states is considered for a polyatomic system. In the adiabatic representation, the n-electronic-state nuclear motion Schrddinger equation is presented along with the structure of the first- and second-derivative nonadiabatic coupling matrices. In this representation, the geometiic phase must be introduced separately and the presence of a gradient term introduces numerical inefficiencies for the solution of that Schrddinger equation, even if [c.214]

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Particle reinforcement is an excellent method for toughening brittle ceramic matrices (3,41,46—48). The toughness imparted to such composites is due to multiple toughening mechanisms including crack deflection, crack pinning, microcracking, residual stress, frictional bridging, particle pullout, and transformation toughening. The mechanisms important to any specific system depend on the physical properties of the particles si2e morphology thermal expansion mismatch with the matrix and strength, toughness, and ductihty. [c.53]

Most computer programs currently used for calculating LEED I-V curves are based on the multiple scattering algorithms outlined in the early works by Pendry, Tong, and van Hove [2.241, 2.242]. The most common way of calculating LEED 7-Vcurves is to subdivide the crystal into atomic layers parallel to the surface. All possible multiple scattering paths inside each layer are first added then combined in a layer diffraction matrix. The total back-diffracted intensity for each LEED spot is then calculated in a second step by combining these layer diffraction matrices in a way that includes all remaining multiple scattering paths between the layers. The amount of computer time needed for calculating a set of I-V curves for one model geometry depends on the number of atoms per surface unit cell and on the number of spots within the LEED pattern. Eor both the dependence is cubic, so the time requirements vary quite substantially from a few seconds up to several hours depending on the complexity of the surface geometry. The computational effort can be significantly reduced by making use of rotational or mirror symmetries at the surface. [c.81]

Combination of 3-D OPLC with multiple development encompasses all of the advantages of three-dimensional and forced-flow planar chromatography, and the separating capacity of multiple development. Favourable conditions could be to start the separation in the first dimension, and then reducing the total solvent strength stepwise at constant mobile phase selectivity to achieve a crude separation on the basis of the polarity of the compounds to be separated. In the second (perpendicular) direction, multiple development could be performed at constant Sj but with variation of the mobile phase selectivity. The third dimension would enable a combination of total solvent strength and mobile phase selectivity for improving the resolution of complex matrices (see Figure 8.14). [c.185]

See pages that mention the term

**Multiplication of matrices**:

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Advanced control engineering (2001) -- [ c.427 ]