Diagonal

About: Diagonal is a research topic. Over the lifetime, 10357 publications have been published within this topic receiving 133454 citations.

Calculating the Singular Values and Pseudo-Inverse of a Matrix

Abstract: A numerically stable and fairly fast scheme is described to compute the unitary matrices U and V which transform a given matrix A into a diagonal form $\Sigma = U^ * AV$, thus exhibiting A’s singular values on $\Sigma $’s diagonal. The scheme first transforms A to a bidiagonal matrix J, then diagonalizes J. The scheme described here is complicated but does not suffer from the computational difficulties which occasionally afflict some previously known methods. Some applications are mentioned, in particular the use of the pseudo-inverse $A^I = V\Sigma ^I U^* $ to solve least squares problems in a way which dampens spurious oscillation and cancellation.

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.