Thread: Linear 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?

Originally Posted by tinng

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. Actually, the general solution depends on n - m arbitrary parameters. How do you find the unique solution with the minimum 2-norm?

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 .