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view toolbox/minij.m @ 0:8f23314345f4 draft
Create local repository for matrix toolboxes. Step #0 done.
author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Wed, 06 May 2015 14:56:53 +0200 |
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function A = minij(n) %MINIJ Symmetric positive definite matrix MIN(i,j). % A = MINIJ(N) is the N-by-N symmetric positive definite matrix with % A(i,j) = MIN(i,j). % Properties, variations: % A has eigenvalues .25*sec^2(r*PI/(2*N+1)), r=1:N, and the eigenvectors % are also known explicitly. % INV(A) is tridiagonal: it is minus the second difference matrix % except its (N,N) element is 1. % 2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and % eigenvalues .5*sec^2((2r-1)PI/4N), r=1:N. % (N+1)*ONES(N)-A also has a tridiagonal inverse. % FLIPUD(TRIW(N,1)) is a square root of A. % References: % J. Fortiana and C. M. Cuadras, A family of matrices, the discretized % Brownian bridge, and distance-based regression, Linear Algebra % Appl., 264 (1997), 173-188. (For the eigensystem of A.) % J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, % Birkhauser, Basel, and Academic Press, New York, 1977, p. 158. % D.E. Rutherford, Some continuant determinants arising in physics and % chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. % (For the eigenvalues of Givens' matrix.) A = min( ones(n,1)*(1:n), (1:n)'*ones(1,n) );