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, 01:21 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, 06:55 AMOpalg
The general solution will be of the form $\displaystyle \{x_0+v:v\in V\}$, where $\displaystyle x_0$ is a particular solution and V is an (n-m)-dimensional subspace. One way to minimise the 2-norm of $\displaystyle x_0+v$ is to construct an orthonormal basis $\displaystyle \{e_1,\ldots,e_{n-m}\}$ of V. The closest point to $\displaystyle x_0$ in V is then $\displaystyle v_0 = \sum_{j=1}^{n-m}\langle x_0,e_j\rangle e_j$, and the solution with smallest 2-norm is $\displaystyle x_0-v_0$.