Transpose

The transpose of a matrix is another matrix, produced by turning rows into columns and vice versa. The transpose of an n by m matrix A is the n by m matrix A defined by 

Process engineering and design using Visual Basic 

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Jl exists in this form only in solution, though stable derivatives of the aldehyde structure are known. The optical antipode of D-glucose in which the positions of every H and OH are transposed is L-glucose. 

This is not the same as (equation Al.4.14). In fact, in this convention, which we can call the S-convention, the multiplication table is the transpose of that given in table Al.4.1. The convention we use and which leads to the multiplication table given in table Al.4.1. will be called the N-convention (where N denotes nuclear-fixed labels). 

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

Thus the transfonnation matrix for the gradient is the inverse transpose of that for the coordinates. In the case of transfonnation from Cartesian displacement coordmates (Ax) to internal coordinates (Aq), the transfonnation is singular becanse the internal coordinates do not specify the six translational and rotational degrees of freedom. One conld angment the internal coordinate set by the latter bnt a simpler approach is to rise the generalized inverse [58]... 

The square brackets denote a vector, and [ ] a transposed vector. The exact expression for the Onsager-Machlup action is now approximated by... 

Here tp denotes the conjugate transpose of ip. Another conserved quantity is the norm of the vector ip, i.e., ip ip = const, due to the unitary propagation of the quantum part. 

Here W is diagonal matrix of singular values, is the transpose of the second re-sultant matrix, being actually the same as the loading matrix in PCA, and X is the matrix, which is applied for further modeling. 

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... 

D is the diagonal matrix containing the eigenvalues A, of S, and U contains the eigenvectors of S. is the transpose of the matrix U. (This expression is often written U SU = D since for real basis functions = U. ) Then the matrix X is given by X = where... 

L has only diagonal entries, the matrix L is identical to its transpose L = L. The atomic Drdinates are thus obtained by multiplying the square roots of the eigenvalues by the jenvectors. 

To obtain the transpose of a tensor T the indices of its components (originally given a.s Tpg) are transposed such that... 

The transpose A of a matrix is obtained by reflecting the matrix through its principal diagonal ... 

If the tr ansformation matr ix is orthogonal, then the tr ansformation is orthogonal. If the elements of A are numbers (as distinct from functions), the transformation is linear. One important characteristic of an orthogonal matrix is that none of its columns is linearly dependent on any other column. If the transfomiation matrix is orthogonal, A exists and is equal to the transpose of A. Because A = A ... 

Find the product A B compare it with the transpose of the product from Problem 6. Deduce a rule for A B. ... 

Is the mahix Q conformable to multiplication into its own transpose What about Q Q What is the dimension of QQ What is the dimension of Q Q ... 

The left side of the normal equations can be seen to be a product including X, its transpose, and m. Matrix multiplication shows that... 

Can a rectangular matrix be both premultiplied and postmultiplied into its own transpose, or must multiplication be either pre- or post- for conformability If multiplication must be either one or the other, which is it ... 

Equation (7-23) is a convenience because it is easier to find the transpose of a large matrix than it is to find its inverse. It is also true that in Huckel theory, A is symmetric, which means that it is equal to its own transpose, leading to the further simplification... 

Recall the definition of a matrix hanspose (section on special mahices in Chapter 2). Transpose the matrices... 

Are the determinants conesponding to the matrix and its transpose equal in these two cases ... 

Program UTMATU This program is designed to read in a real matrix. A, a real transformation matrix, B, perform the transformation X = B(transpose) A B, and output the result. With the program limitations deseribed above, UTMATU memory usage is 1960040 bytes. 

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