How can QR decomposition be used to solve least squares problems?

[epkid08]

Homework Statement

Okay so I'm supposed to find the least squares solution of a set of equations, which I can do, but it adds that I must use QR decomposition. I don't really know how to apply QR decomposition to this problem.

Problem: Find the least squares solution of
[tex]x_1 + x_2 = 4[/tex]
[tex]2x_1+x_2 = -2[/tex]
[tex]x_1 - x_2 = 1[/tex]

Use your answer to find the point on the plane spanned by (1,1,2) and (1,-3,1) that is closest to (1,4,3).

Homework Equations

The Attempt at a Solution

 

[tjackson3]

Forget the least squares stuff for a moment. Find the QR factorization of A, so we have A = QR. For least squares, you need to solve the equation [itex]A^TAx = A^Tb[/itex] right? So substitute in A = QR to get [itex](QR)^T(QR) = R^TQ^TQR[/itex]. The whole gimmick of orthonormal matrices such as Q is that [itex]Q^TQ = I[/itex], so we have [itex]A^TAx = R^TRx = (QR)^Tb[/itex]. So [itex]R^TRx= R^TQ^Tb[/itex]. sp the least squares equation comes down to [itex]Rx = Q^Tb[/itex]. But R is invertible, so the least squares solution is just

[tex]x = R^{-1}Q^Tb[/tex]