Matrices transposed

The transpose of a square matrix is, of course, another square matrix. The transpose of a symmetric matrix is itself. One particularly important transpose matrix is the adjoint natris, adJA, which is the transpose matrix of cofactors. For example, the matrix of cofactors ul liie 3x3 matrix... 

The transpose matrix of a matrix A is obtained by interchanging the rows and columns of A. If the matrix A is given by equation (1.1), then its transpose is... 

The value of a determinant is unchanged if the rows are written as columns. Thus, the determinants of a matrix A and its transpose matrix are equal. 

For orientation measurements, this tensor also needs to be expressed in the coordinate system OXYZ, axrz, using the matrix transformation u.xyz = Oaxyz / where O is a matrix whose elements are the direction cosines of the coordinate axes and is its transposed matrix [44]. 

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... 

The transposed matrix A1 is defined as the interchange of rows and columns of A. This can also be seen as the reflection of all elements of A at its main diagonal (along ay, i=j), according to... 

Note that on a finite-dimensional vector space V, a linear operator is Her-mitian if and only if T = T. More concretely, in C", a linear operator is Hermitian-symmetric if and only if its matrix M in the standard basis satis-lies M = M, where M denotes the conjugate transpose matrix. To check that a hnear operator is Hermitian, it suffices to check Equation 3.2 on basis vectors. Physics textbooks often contain expressions such as (+z H — z). These expressions are well defined only if H is a Hermitian operator. If H yNQK not Hermitian, the value of the expression would depend on where one applies the H. 

Here a denotes the typical element of the transposed matrix a. ... 

Here UT denotes the transpose matrix, namely if U = (uiyj) is given by its entries, then UT = (Ujyi) with rows and columns exchanged, and U = (ujyi) is the complex conjugate matrix with complex conjugate transposed entries. 

Hint Make use of the definition of the inverse matrix, eq. (7), and the property of the transposed matrix.]... 

Here vT is the transposed matrix of stoichiometric numbers and Tint is the matrix of stoichiometric coefficients for intermediates. Elements of the latter are taken to be negative if substance is consumed in a given reaction step, positive if it is formed, and zero if substance is not involved in the reaction step. Multiplication of matrix vT (P-by-s) by matrix Tmt (s-by-/tot) gives the matrix vTrint whose size is (P-by-/tot) (s is the number of steps). 

Here vT is the transposed matrix of the Horiuti numbers (stoichiometric numbers) and Tint the matrix of the intermediate stoichiometric coefficients. The size for the matrices vT and rint is (P x S) and (S x Jtot), respectively, where S is the number of steps, Jtot the total number of independent intermediates, and P the number of routes. Due to the existence of a conservation law (at least one), the catalyst quantity and the number of linearly independent intermediates will be... 

Show also that the inverse matrix E 1 is equal to the transpose matrix ET (where rows and columns are interchanged) ... 

X = X a, one should have the similarity transformation a-1 T a, whereas the transposed matrix has the related special property... 

It should be observed that the classical canonical form A is by no means symmetric, except in the special case when it happens to be diagonal. The transposed matrix A has the same elements as A on the diagonal, but the Os and Is are now on the line one step below the diagonal. For a specific Jordan block of order p, one obtains from Eq. (A.3)... 

As this point it is important to note that the B matrix is usually not square and therefore cannot be inverted. In fact, it transforms from Cartesian (dimension 3n) to internal coordinates (dimension 3n - 6). In simple cases, six dummy coordinates (Tx, Ty, Tz, Rx, Ry, Rz) may be added to the 3 - 6 internal ones in order to obtain in invertible square matrix. However, in some cases the symmetry of the problem makes it necessary to introduce redundant non-linearly independent coordinates (6 CCC angles for benzene or 6 HCH angles for CHq). Gussoni et al. (1975) has shown that it is possible to use the transposed matrix instead of the inverse one and that this choice is the only one which ensures invariance of the potential energy upon coordinate transformation. We can therefore write... 

The first step in this discussion concerns the presentation of the matrix of the independent variables (X), the experimental observation vector of the dependent variable (Y) and the column matrix of the coefficients (B) as well as the transposed matrix of the independent variables (X. All these terms are introduced by relation (5.84). A fictive variable Xq, which takes the permanent value of 1, has been considered in the matrix of the independent variables ... 

The particularization of the system of the normal equations (5.9) into an equivalent form of the relationship between the process variables (5.83), results in the system of equations (5.85). In matrix forms, the system can be represented by relation (5.86), and the matrix of the coefficients is given by relation (5.87). According to the inversion formula for a matrix, we obtain the elements for the inverse matrix of the matrix multiplication (XX ), where (X ) is the transpose matrix of the matrix of independent variables. 

Jezierska et al. [67] applied the Kohonen neural network to select the most relevant descriptors. Here a Kohonen network is built with the transposed matrix, i.e., with the matrix where the roles of descriptors and molecules are exchanged. From the map of descriptors, 36 descriptors were selected. This number of descriptors was further reduced to six, five, four, or three descriptors. Statistical parameters of compared models are reported in Table 3 of [67]. It is evident from the table that the model built with four selected descriptors show comparable parameters to the model built with 36 descriptors. The selected descriptors belong to topostructural and topochemical classes. 

The prime means here the transposed matrix. Explicitly in matrix elements, L = /y and L = /ly, the last two equations read... 

Note that the normal system can be obtained formally by multiplication of the original system (3.2) by the transposed matrix AT. However, in general cases, the pseudosolution mo is not equivalent to the solution of the original system, because the new system described by equation (3.8) is not equivalent to the original system (3.2) if matrix A is not square. The main characteristic of the pseudo-solution is that it provides the minimum of the misfit functional. 

Thus, each row of matrix A, each element of vector y (see Eq. 5.32) and each column of the transpose matrix (see Eq. 5.37) is changed by the multiplier that is inversely proportional to the square root of the experimental error in the corresponding experimental data point. Alternatively, the weighted least squares solution may be expressed as follows... 

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