Product of a matrix and

C CT] is known as the pseudo inverse of C. Since the product of a matrix and its inverse is the identity matrix, [C CT][C CT] disappears from the right-hand side of equation [32] leaving... 

The product of a matrix and a scalar is defined to be the matrix formed by multiplying each element of the original matrix by the scalar. For... 

The product of a matrix and its inverse is commutative and equals a unit matrix... 

Here stands for a column vector, and on the ihs of (4.3.10) we have a product of a matrix and a column vector. Since P is symmetric, the right and the left eigenvectors that correspond to A/ are identical. 

If a square matrix has an inverse, the product of the matrix and its inverse equals the unit matrix. The inverse of a matrix A is denoted by A1. 

Some analytical instruments produce a table of raw data which need to be processed into the analytical result. Hyphenated measurement devices, such as HPLC linked to a diode array detector (DAD), form an important class of such instruments. In the particular case of HPLC-DAD, data tables are obtained consisting of spectra measured at several elution times. The rows represent the spectra and the columns are chromatograms detected at a particular wavelength. Consequently, rows and columns of the data table have a physical meaning. Because the data table X can be considered to be a product of a matrix C containing the concentration profiles and a matrix S containing the pure (but often unknown) spectra, we call such a table bilinear. The order of the rows in this data table corresponds to the order of the elution of the compounds from the analytical column. Each row corresponds to a particular elution time. Such bilinear data tables are therefore called ordered data tables. Trilinear data tables are obtained from LC-detectors which produce a matrix of data at any instance during the... 

The product of a matrix with a diagonal matrix is used to multiply the rows or the columns of a matrix with given constants. If X is an nxp matrix and if D is a diagonal matrix of dimension n we obtain a product Y in which the ith row y, equals the ith row of X, i.e. x, multiplied by the ith element on the main diagonal of... 

That was the hard part. It now remains to calculate out the expressions shown in equation 4-10, to find the final values for the unknowns in the original simultaneous equations. Thus, we need to form the matrix product of [A]-1 and [C] ... 

If ns absorbance measurements, y. ..yns, are taken for ns different mixtures of the nc components, then ns equations of the kind (3.8) can be written. Again, it is possible and more convenient to use the vector/matrix notation the vector y is the product of a matrix C that contains, as rows, the concentrations of the components in each solution, multiplied by the same column vector a with the molar absorptivities. 

The product of a number and a matrix is another matrix obtained by multiplying each of the elements of the matrix by the number. For example. 

S v are elements of the overlap matrix. Similar types of expressions may be constructed for density functional and correlated models, as well as for semi-empirical models. The important point is that it is possible to equate the total number of electrons in a molecule to a sum of products of density matrix and overlap matrix elements. ... 

Next we show that this integral is invariant under group multiplication on the left. Recall from Section 4.2 that SU(2) is isomorphic to the unit quaternions. From Exercise 4,25 we know that multiplication of a unit quaternion q on the left by a unit quaternion qo corresponds to the product of a matrix in 5(9(4) (corresponding to qo) and a vector in 5- C (corresponding to q). See Figure 6.3. 

For tridiagonal matrices, the decomposition of the matrix into a product of a lower and an upper diagonal matrix leads to an efficient algorithm known as the Thomas algorithm. For a system of the form... 

In conclusion, we note that thus far we have derived matrix elements of the transformed Hamiltonian Xfor a given block in the complete matrix labelled by a particular value of rj rather than an effective Hamiltonian operating only within the subspace of the state rj. It is an easy matter to cast our results in the form of an effective Hamiltonian for any particular case since the matrix elements involved in either the commutator bracket formulation (contact transformation) or the explicit matrix element formulation (Van Vleck transformation) can always be factorised into a product of a matrix element of operators involved in X associated with the quantum number rj and a matrix element of operators that act only within the subspace levels of a given rj state, associated with the quantum number i. This follows because the basis set can be factorised as in equation (7.47). The matrix element involving the rj quantum number can then either be evaluated or included as a parameter to be determined experimentally, while the... 

The bracket (bra-c-ket) in (

component vectors. This notation was introduced in Section 3.2 as a shorthand for the scalar product integral. The scalar product of a ket ip) with its corresponding bra (y> gives a real, positive number and is the analog of multiplying a complex number by its complex conjugate. The scalar product of a bra ( / and the ket Axpt) is expressed in Dirac notation as [y>j A ipi) or as (j A i). These scalar products are also known as the matrix elements of A and are sometimes denoted by Ay. 

Thus, if A solves an eigenvalue equation of the form (E2.8), then a coordinate transformation can be found which will yield a diagonalized form of the matrix. The matrix A is itself a tensor (see Section E5.2) which can be constructed by taking the product of a column and a row vector... 

Let X be the mean centred (and, if necessary, scaled) (N xK) data matrix. It was seen that if the variance-covariance matrix (correlation matrix) X X has K distinct eigenvalues it is possible to factorize X as the product of a matrix of object scores, T, and the transpose matrix, F, of the eigenvectors. 

The outer product of a row and column matrix, both of order m, is a square matrix of order m and is written as [C] = with... 

Two matrices can be added only if they have the same number of rows and the same number of columns. The product of a scalar and a matrix is defined by... 

Thus the vector of the differences of concentrations is a product of a matrix for the general stoichiometric coefficients and the vector of the linear independent degrees of advancements. This relationship is valid for each time t,. For various times given as f (/ = 0,1,2,...,/n) these measured differences in concentration can be arranged in a matrix... 

The matrix Y) nspectra nlam) is the product of a matrix of concentrations C nspectra ncomp) and a matrix A(ncomp nlam). C contains as columns the concentration profiles of the reacting components and the matrix A contains, as rows, their molar absorption spectra. 

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