Main diagonal

About: Main diagonal is a research topic. Over the lifetime, 872 publications have been published within this topic receiving 10448 citations.

Some necessary conditions for common-factor analysis

Abstract: LetR be any correlation matrix of ordern, with unity as each main diagonal element. Common-factor analysis, in the Spearman-Thurstone sense, seeks a diagonal matrixU 2 such thatG = R − U 2 is Gramian and of minimum rankr. Lets 1 be the number of latent roots ofR which are greater than or equal to unity. Then it is proved here thatr ≧s 1. Two further lower bounds tor are also established that are better thans 1. Simple computing procedures are shown for all three lower bounds that avoid any calculations of latent roots. It is proved further that there are many cases where the rank of all diagonal-free submatrices inR is small, but the minimum rankr for a GramianG is nevertheless very large compared withn. Heuristic criteria are given for testing the hypothesis that a finiter exists for the infinite universe of content from which the sample ofn observed variables is selected; in many cases, the Spearman-Thurstone type of multiple common-factor structure cannot hold.

On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix

Abstract: We consider bipartite matching algorithms for computing permutations of a sparse matrix so that the diagonal of the permuted matrix has entries of large absolute value. We discuss various strategies for this and consider their implementation as computer codes. We also consider scaling techniques to further increase the relative values of the diagonal entries. Numerical experiments show the effect of the reorderings and the scaling on the solution of sparse equations by a direct method and by preconditioned iterative techniques.

The cyclic Jacobi method for computing the principal values of a complex matrix

Abstract: is diagonal (T denotes the transpose), then the main diagonal of A is made up of the numbers Xi in some order. If it is desired to compute the Xi numerically, this result is of no immediate use, since for n> 2 there exists no manageable expression for the general orthogonal matrix of order n. However, Jacobi [6] suggested the computation of the set of Xi as the limiting set of diagonal elements of a sequence of matrices which are generated from A recursively by plane rotations. For k = 0, 1, 2, * *, let ck =cf)k be a real angle and (i, j) = (ik, jk), a pair of integers such that 1