George Tuma 122 Report post Posted August 31, 2005 What can I say about solution of the Ax=b matrix equation, where A is a semipositive definite matrix? Thank for help 0 Share this post Link to post Share on other sites
Winograd 380 Report post Posted August 31, 2005 I can say that this is homework.. 0 Share this post Link to post Share on other sites
Eelco 301 Report post Posted August 31, 2005 that you could semisolve it with conjugate gradients?dunno.whats semipositive anyway? 0 Share this post Link to post Share on other sites
George Tuma 122 Report post Posted August 31, 2005 I appologize ...exchange "positive semidefinite" for "semipositive definite" 0 Share this post Link to post Share on other sites
Eelco 301 Report post Posted August 31, 2005 Quote:A real or Hermitian square matrix A is positive semi-definite or non-negative definite if xAx >=0 for all non-zero x.but that doesnt really say anything about the solution.all i know about the solution is that you can find it using the conjugate gradients method, but thats about it. 0 Share this post Link to post Share on other sites
George Tuma 122 Report post Posted August 31, 2005 I am not an expert in linear algebra. It is only a question how to comment some procedure. For example a problem to find the minimum value of the objective function J(x)=x’(kA’A+I) x-y’x-x’y+y’y leads to the formulae x=inv(kA’A+I)*y (x,y are vectores, A is a rectangular matrix). The matrix kA’A+I is square symmetric and positive definite.More general problem to minimize J(x1,x2)= k1 x1’A1’A1x1+ k2 x2’A2’A2x2+(y’-x1’-x2’) (y-x1-x2) gives a system matrix which is only positive definite. What can I say about the matrix solution? From the theoretical point of view, solution exists or not? I prepared a script for Matlab based on the PCG method for x1,x2,…,xP (arbitrary P) and it works.Best regardsGeorge 0 Share this post Link to post Share on other sites
George Tuma 122 Report post Posted August 31, 2005 A mistake:More general problem to minimize J(x1,x2)= k1 x1’A1’A1x1+ k2 x2’A2’A2x2+(y’-x1’-x2’) (y-x1-x2) gives a system matrix which is only positive semidefinite. 0 Share this post Link to post Share on other sites
Eelco 301 Report post Posted August 31, 2005 i dont have a clue what that notation with all those ' means. transpose maybe? 0 Share this post Link to post Share on other sites
Eelco 301 Report post Posted August 31, 2005 btw the result of an outer product is always a positive definite matrix. 0 Share this post Link to post Share on other sites
George Tuma 122 Report post Posted August 31, 2005 No, the resulting matrix A'A is only positive semidefinite for any rectangular matrix A (A' is transpose of A). You are right only for the square regular matrix A. 0 Share this post Link to post Share on other sites
LilBudyWizer 491 Report post Posted September 1, 2005 Well, it lets you say certain methods are valid to use to solve it numerically. 0 Share this post Link to post Share on other sites