Riley hobson bence · Problem 8.42

Matrices and vector spaces

Mathematics — Methods · singular-value-decomposition

Problem

Find the SVD form of the matrix

A = \begin{pmatrix} 22 & 28 & -22 \\ 1 & -2 & -19 \\ 19 & -2 & -1 \\ -6 & 12 & 6 \end{pmatrix}.

Use it to determine the best solution

\mathbf{x}

of the equation

A\mathbf{x} = \mathbf{b}

when (i)

\mathbf{b} = (6\ -39\ 15\ 18)^T

, (ii)

\mathbf{b} = (9\ -42\ 15\ 15)^T

, showing that (i) has an exact solution, but that the best solution to (ii) has a residual of

\sqrt{18}