Permutation matrices

Figure 2-80. The permutation matrix oFthe reference isomer the second line gives the indices of the sites of the skeleton and the first line the indices of the ligands (e.g.. the ligand with index 3 is on skeleton site 3).

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),...

Figure 2-85. The permutation matrix of the reference isomer for double bonds.

One of the segments represents the reference isomer, and the permutation matrix of this one gets the descriptor +1) (Figure 2-85). 

LV Factorization of a Matrix To eveiy m X n matrix A there exists a permutation matrix P, a lower triangular matrix L with unit diagonal elements, and a.nm X n (upper triangular) echelon matrix U such that PA = LU. The Gauss elimination is in essence an algorithm to determine U, P, and L. The permutation matrix P may be needed since it may be necessaiy in carrying out the Gauss elimination to... 

Returning to our reconciliation problem, the Q-R decomposition of matrix A2 allows us to obtain Qu and Ru matrices and the permutation matrix IIU, such that... 

Remark 2. The permutation matrix IIU, obtained as a by-product of the Q-R factorization procedure of A2, enables an easy classification of the unmeasured process variables, as is indicated by Eq. (4.15). The variables in subset un ru correspond to the minimum number and the location of measurements needed for the system to satisfy the estimability condition, that is, that all unmeasured variables be determinable. 

In this section we restrict considerations to an nxn nansingular matrix A. As shown in Section 1.1, the Gauss-Jordan elimination translates A into the identity matrix I. Selecting off-diagonal pivots we interchange some rows of I, and obtain a permutation matrix P instead, with exactly one element 1 in each row and in each column, all the other entries beeing zero. Matrix P is called permutation matrix, since the operation PA will interchange some rows of A. ... 

The Gaussian elimination also enables us to decompose the matrix in (1.46). We already have the upper triangular in (1.49). To form the permutation matrix P we will interchange those rows of the identity matrix I that have been interchanged in A in the course of the Gaussian elimination. Let (i,k ) denote the operation of interchanging rows i and k in the i-th step, then what we did is (1,1), (2,4) and (3,4). These operations applied to the identity matrix I result in the permutation matrix... 

The decomposition is "in place", i.e., all results are stored in the locations that matrix A used to occupy. The upper triangular matrix U will replace the diagonal elements of A and the ones above, whereas L is stored in the part below the diagonal, the unit elements of its diagonal being not stored. We will then say that the matrices are in "packed" form. The permutation matrix P is not stored at all. As discussed, the row interchanges can be described by n-l pairs (i,k ), and all information is contained in a... 

B is called a permutation matrix. Each column of B, say, b is a column of an identity matrix. The /th column of the matrix product AB is A b, which is the /th column of A. Therefore, post multiplication of A by B simply rearranges (permutes) the columns of A (hence the name). Each row of the product BA is one of the rows of A, so the product BA is a rearrangement of the rows of A. Of course, A need not be square for us to permute its rows or columns. If not, the applicable pennutation matrix will be of different orders for the rows and columns. 

Prove that exchanging two columns of a square matrix reverses the sign of its detenninant. (Hint use a permutation matrix. See Exercise 6.)... 

Exchanging the first two columns of a matrix is equivalent to postmultiplying it by a permutation matrix B = [e2,ei,e3,e4,...] where e, is the /th column of an identity matrix. Thus, the detenninant ofthe matrix is AB = A B. The question turns on the determinant of B. Assume that A and B have n columns. To obtain the detenninant of B, merely expand it along the first row. The only nonzero term in the determinant is (-1 ) I i = -1, where I i is the (n-1) x (n-1) identity matrix. This completes the proof. 

Every group is isomorphous to some permutation group. It is easy to find a representation of a permutation group by using a permutation matrix. Each row and column of such a matrix has but one non-zero element and that is unity. The row and column thus designate the initial and final locations of the object permuted. 

In eq. (4) the cosets themselves are used as a basis for G, and from eq. (3) gs H is transformed into gr H by gj. Since the operator gy simply re-orders the basis, each matrix representation in the ground representation Tg is a permutation matrix (Appendix A1.2). Thus the 5th column of T8 has only one non-zero element,... 

A permutation pseudo-permutation) matrix is one in which every element in each row and column is equal to zero, except for one element which is +1 (—1 or +1). For convenience of reference the defining relations for special matrices are summarized in Table ALL... 

The last equation expresses the fact that the set of distances is mapped by p onto itself, therefore the matrix r -r y (F) is a permutation matrix of dimension /K ... 

Thereby the matrix 11(F) denotes a K-dimensional permutation matrix and F (F) a 3 by 3 orthogonal matrix. The form of this representation follows from the fact that each isometric transformation maps the NC Xk, Zk, Mk onto a NC which by definition has the same set of distances, i.e. is isometric to NC Xk, Zk, Mk. Expressed alternatively, the nuclear configurations NC Xk( ), Zk, Mk and NC Xk(F 1 ( )), Zk, Mk are properly or improperly congruent up to permutations of nuclei with equal charge and mass for any F G ( ). The set of matrices Eq. (2.12) forms a representation of J d) by linear transformations and will furtheron be denoted by... 

For a set of equivalent nuclei in general site the matrices 11(G) are identical with the right regular representation matrices2 lf the nuclear position vectors of all K nuclei of a SRM are included in the basis Xkd), 11(G) denotes a K by K permutation matrix. In addition to the matrix groups (2.49) and (2.49 ) the set... 

The first factor of the direct products denotes a 8 by 8 permutation matrix. If the position vectors are arranged in the order... 

To make ( (s) diagonally dominant, it is necessary to select a specific type of controller. Rosenbrock has proposed that K(s) K.qKh(s)Kr (s), i.e., a product of three controller matrices. Ka is a permutation matrix, which scales the elements in G(s)K(s) and makes some preliminary assignment of single loop connections, usually to assure integrity. This step can be used to make (G(s)K(s)diagonally dominant as s 0. K s) can be chosen to meet stability criteria. Finally the elements of Kc(s), a diagonal matrix, can be selected to improve performance of the system. The proper selection of and Kt>(s) are the most difficult parts of the design process, and this step should be considered iterative, especially for an inexperienced user. 

In the PDS algorithm, a small set of calibration-transfer samples are measured on a primary instrument and a secondary instrument, producing spectral response matrices X, and X2. A permutation matrix F (Procrustes transfer matrix) is used to map spectra measured on the secondary instrument so that they match the spectra measured on the primary instrument. 

It is now easily checked that, if P, is a permutation matrix which reverses the order of the basic element associated with this specific Jordan block, so that... 

If, finally, P is the permutation matrix built up in block-diagonal form from all the submatrices Pt associated with the various Jordan blocks, one... 

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