Main diagonal

A triangular matrix is a matrix all of whose elements above or below the main diagonal (set of elements an,. . . , a j) are zero. 

A diagonal matrix is one such that all elements both above and below the main diagonal are zero (i.e., ay = 0 for all i J). If all diagonal elements are equal, the matrix is called scalar. If A is diagonal, A =... 

If A is a square matrix and if principal submatrices of A are all nonsingular, then we may choose P as the identity in the preceding factorization and obtain A = LU. This factorization is unique if L is normahzed (as assumed previously), so that it has unit elements on the main diagonal. 

A tridiagonal matrix is one in which the only nonzero entries he on the main diagonal and the diagonal just above andjust below the main diagonal. The set of equations can be written as... 

For a square matrix, the principal or main diagonal goes from the upper left-hand comer to the lower right-hand comer of the matrix. Thus, the principal diagonal has elements Ajj. A symmetric (square) matrix has elements that are symmetric about the principal diagonal, that is... 

Dense (few zero elements) and small. A banded matrix has all zero elements except for a band centered on the main diagonal, e.g.,... 

For general matrices the reduction by triangular matrices requires less computation and is probably to be preferred. But if A is hermi-tian, observe that the use of the unitary reduction produces a matrix H that is again hermitian, hence, that is tridiagonal in form, having zeros everywhere except along, just above, and just below the main diagonal. 

In general, Q is a nonsymmetrical matrix whose components off the main diagonal are the coupling coefficients among the various fluxes involved. 

Let A be new tridiagonal matrices differing from A solely by the zero elements on the main diagonals. While solving equation (4) and equation (5) we should save in the storage the vectors 4 /2n+i 2/2tj+2>... 

Such a triangle additive scheme will be economical once we involve economical diagonal operators a = 1,2,..., m. Economical schemes arising in practical implementations of multidimensional mathematical-physics problems turn out to be triangle additive schemes (usually lower, but sometimes upper), whose matrices are of a special structure. As a rule, nonzero elements of the matrix (C ap) stand only on one or two diagonals adjacent to the main diagonal. With this in mind, the scheme... 

The adjacency matrix A(G) of a molecular graph G with N vertices is the square NxN symmetric matrix in which [A],j=l if vertex Vj is adjacent to vertex Vj and [A],j=0 otherwise. The adjacency matrix is symmetric, with all elements on the main diagonal equal to zero. The sum of entries over row i or column i in A(G) is the degree of vertex Vj, 5j. Usually, the adjacency matrix is based on weighted molecular graphs in which heteroatoms are represented as vertex parameters and... 

In a square nxn matrix A, the main diagonal or principal diagonal consists of the elements for all i ranging from 1 to n. The latter are called the diagonal elements all other elements are off-diagonal. 

The trace of a square matrix A of dimension n is equal to the sum of the n elements on the main diagonal ... 

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... 

From the spectral decomposition we can deduce that A and A always have the same rank, since the rank of a matrix is equal to the number of nonzero eigenvalues which are also the elements on the main diagonal of A (or A )-... 

The matrix Cp contains the variances of the columns of X on the main diagonal and the covariances between the columns in the off-diagonal positions (see also Section 9.3.2.4.4). The correlation matrix Rp is derived from the column-standardized matrix Zp ... 

Equation (31.3) defines the eigenvalue decomposition (EVD), also referred to as spectral decomposition, of a square symmetric matrix. The orthonormal matrices U and V are the same as those defined above with SVD, apart from the algebraic sign of the columns. As pointed out already in Section 17.6.1, the diagonal matrix can be derived from A simply by squaring the elements on the main diagonal of A. 

Sometimes it is claimed that the double-centered biplot of latent variables 1 and 2 is identical to the column-centered biplot of latent variables 2 and 3. This is only the case when the first latent variable coincides with the main diagonal of the data space (i.e. the line that makes equal angles with all coordinate axes). In the present application of chromatographic data this is certainly not the case and the results are different. Note that projection of the compounds upon the main diagonal produces the size component. 

Distances in are different from those in the usual space S. A weighted space can be represented graphically by means of stretched coordinate axes [2]. The latter result when the basis vectors of the space are scaled by means of the corresponding quantities in Vw, where the vector w contains the main diagonal elements of W. Figure 32.3 shows that a circle is deformed into an ellipse if one passes from usual coordinate axes in the usual metric I to stretched coordinate axes in the weighted metric W. In this example, the horizontal axis in 5, is stretched by a factor. l-6 = 1.26 and the vertical axis is shrunk by a factor Vo.4 = 0.63. 

In Section 29.3 it has been shown that a matrix generates two dual spaces a row-space S" in which the p columns of the matrix are represented as a pattern P , and a column-space S in which the n rows are represented as a pattern P". Separate weighted metrics for row-space and column-space can be defined by the corresponding metric matrices and W. This results into the complementary weighted spaces and S, each of which can be represented by stretched coordinate axes using the stretching factors in -J v and, where the vectors w and Wp contain the main diagonal elements of W and W. ... 

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