Let A be m x n, (m<n) and of full rank (ie rank(A) = m). Then, the linear system Ax = b has infinite many solutions. Acutally, the general solution depends on n - m arbitrary parameters. How do you find the unique soultion iwth the minumum 2-norm?

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- Oct 19th 2008, 02:21 AMtinngLinear system
Let A be m x n, (m<n) and of full rank (ie rank(A) = m). Then, the linear system Ax = b has infinite many solutions. Acutally, the general solution depends on n - m arbitrary parameters. How do you find the unique soultion iwth the minumum 2-norm?

- Oct 19th 2008, 07:55 AMOpalg
The general solution will be of the form , where is a particular solution and V is an (n-m)-dimensional subspace. One way to minimise the 2-norm of is to construct an orthonormal basis of V. The closest point to in V is then , and the solution with smallest 2-norm is .