Matrices transpositions

Figure 2-3 shows an example spreadsheet. Cells A3 C4 contain the elements of the 2x3 matrix A. In order to perform the matrix transposition cells E3 F5, which will contain the result, A1, have to be pre-selected. Next =TRANSPOSE(A3 C4) is typed on the Excel command line followed by the SHIFT+CTRL+ENTERl key combination. 

Figure 2-4. Matrix transposition via Excels interactive graphical user interface...

In Excel, the cells comprising the prospective result Y have to be pre-selected as we have already seen for matrix transposition. For this, we need to predict the dimensions of Y from the row dimension of C and column dimension of A. Also, there is no direct operator for matrix multiplication in Excel. The function MMUUI in conjunction with the SHIFT+CTRL+ENTER key... 

In Excel, matrix inversion can be performed similarly to matrix transposition (see earlier). Figure 2-13 gives an example. Cells D3 E4, defining the target matrix, have to be pre-selected and now the MINVERSE function is applied to the source cells A3 B4. Finally, the SHIFT+CTRL+ENTER key combination is used to confirm the matrix operation. 

Here, or e S is an element of the permutation group of the n electron labels, and sgn(a) is its parity. Equation (6.43) indicates that this permutation can equally well be applied to the component labels, since the determinant is invariant under matrix transposition. We can now calculate the matrix element in the symmetry operator ... 

Here ( ) denotes complex conjugation, ( ) denotes matrix transposition, and ( ) denotes differentiation with respect to s. This equation shows that if the complex eigensolutions sj and u, can be... 

The superscript T indicates matrix transposition. S = FF is the two-point correlation matrix of the basis vectors only this parameter appears in the solution and not the basis vectors themselves. The nonlinearity of the problem is taken into account through iterative application of Eq. (8.2.8). The error covariance matrix for the retrieval temperature profile due to instrument noise propagation is... 

In the actual 2D procedure, the first interferogram is formed from the first data point from each of the original spectra, and the second interferogram is formed from the second data point from each of the original spectra. Mathematically, this process is a simple matrix transposition. In fact, if these spectra are thought of as the rows of the data matrix S(tc, Fa) then the columns represent the variation of the intensity of a given frequency component Fa as a function of r. The result of this complete process is a 2D frequency map, on which the data are plotted in the form of a series of spectra (stacked or white-washed plot) or in the form of a contour plot in which each successive contour represents a higher intensity. 

In the example, we proceed from the right (rotated molecule) to the left (reference molecule). The first transposition is done with ligands 4 and 2, in order to obtain the fir.st part of the reference. scqttencc 1 2". Then, only a pcrmtttation of ligands 3 and 4 has to be done to obtain the reference matrix on the left-hand side. Thus, in total, we have executed two transpositions (4 2) and (3 4),... 

Figure 2-81. The permutation matrices oftwo structures that differ through rotation by 120 T The permutation matrix of the rotated Isomer can be brought Into correspondence with the permutation matrix of the reference isomer by two Interchanges of two ligands (transpositions),...

Transposition The matrix obtained from A by interchanging the rows and columns of A is called the transpose of A, written A or A. ... 

The superscript t denotes a transposition of the r-vector, i.e. converting it from a column to a row vector. The rr notation for the quadmpole moment therefore indicates a 3 x 3 matrix containing the products of the x-, y- and z-coordinates, e.g. the Qxy component is calculated as the expectation value of xy. 

There are actually two independent time periods involved, t and t. The time period ti after the application of the first pulse is incremented systematically, and separate FIDs are obtained at each value of t. The second time period, represents the detection period and it is kept constant. The first set of Fourier transformations (of rows) yields frequency-domain spectra, as in the ID experiment. When these frequency-domain spectra are stacked together (data transposition), a new data matrix, or pseudo-FID, is obtained, S(absorption-mode signals are modulated in amplitude as a function of t. It is therefore necessary to carry out second Fourier transformation to convert this pseudo FID to frequency domain spectra. The second set of Fourier transformations (across columns) on S (/j, F. produces a two-dimensional spectrum S F, F ). This represents a general procedure for obtaining 2D spectra. 

Since the individual coordinate transformations T depend continuously and differentially on some rotation angles specifying these transformations, the same must hold for the combined transformations, Xk > as well, since transposition and matrix... 

The transpose of a matrix or a vector is formed by assembling the elements of the first row of the matrix as the elements of the first column of the transposed matrix, the second row into the second column, and so on. In other words, atj in the original matrix A becomes the component aji in the transpose Ar. Note that the position of the diagonal components atj) are unchanged by transposition. If the dimension of A is n X m, the dimension of Ar is m X n (m rows and n columns). If square matrices A and AT are identical, A is called a symmetric matrix. The transpose of a vector x is a row... 

It is helpful to distinguish matrices, vectors, scalars and indices by typographic conventions. Matrices are denoted in boldface capital characters (A), vectors in boldface lowercase (a) and scalars in lowercase italic characters (s). For indices, lower case characters are used (i). The symbol t indicates matrix and vector transposition (A4, a4). 

In the alternate computation of the best C from A, the same function can be used. It computes the rows of C in an analogue loop, using the appropriate row of Y and the complete matrix A. The computation of positive rows of C, using Isqnonneg requires the appropriate transpositions for the rows of C and Y and the matrix A. 

Now we may find the matrix representation, U, of the operators. The dimensions of the matrices will be the same as the dimensions of the irreducible representation used. The matrix representation of the identity operator, U E), will of course be the identity matrix. If it is noted that any permutation may be written as a product of transpositions (permutations of order 2), and any transposition may be written as a product of elementary transpositions p p + 1) [74], then it is only nessesary to find matrix representations of the elementary transpositions. The diagonal elements of the elementary transposition p p + 1) are given by... 

Let us recall that the transformation matrix D1 is the Hermitian conjugate to matrix D and obtained from D by its transposition (D) and complex conjugation ( > ). The unitarity of the matrix implies that... 

The symmetry properties of the quantities used in the theory of complex atomic spectra made it possible to establish new important relationships and, in a number of cases, to simplify markedly the mathematical procedures and expressions, or, at least, to check the numerical results obtained. For one shell of equivalent electrons the best known property of this kind is the symmetry between the states belonging to partially and almost filled shells (complementary shells). Using the second-quantization and quasispin methods we can generalize these relationships and represent them as recurrence relations between respective quantities (CFP, matrix elements of irreducible tensors or operators of physical quantities) describing the configurations with different numbers of electrons but with the same sets of other quantum numbers. Another property of this kind is the symmetry of the quantities under transpositions of the quantum numbers of spin and quasispin. 

In eqs. (7a) and (7b) v) is a matrix of one column containing the components of v, and u is a matrix of one row, which is the transpose of w ), the matrix of one column containing the components of u, complex conjugated. In eq. (6), transposition is necessary to conform with the matrix representation of the scalar product so that the row x column law of matrix multiplication may be applied. Complex conjugation is necessary to ensure that the length of a vector v... 

The matrix elements in both BRe and Bim will, for both uniform and exterior complex scaling, be built from terms which, when the matrix representation of the rotated operator is constructed, are multiplied with complex constants. This will make the matrices BRe and BIm complex, but they will still be symmetric and antisymmetric, respectively, with respect to transposition, i.e.,... 

Transposition. In Mathematica the Transpose[a] transposes the first two levels of a. Equations 5.1-14 and 5.1-23 give a matrix and its transpose. 

The transposition to the above algorithm to the COSMO framework is straightforward. On the other hand, the extention to IEF-type methods require some attention. Indeed, a direct transposition of the above algorithm to the IEFPCM framework leads to the matrix elements... 

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