Chapter 4.6, Problem 32E

Rank 1 matrices are important in some computer algorithms and several theoretical contexts, including the singular value decomposition in Chapter 7. It can be shown that an m × n matrix A has rank 1 if and only if it is an outer product; that is, A = uv^T for some u in ℝ^m and v in ℝⁿ. Exercises 31-33 suggest why this property is true.

32. Let $u=\left[\begin{array}{c}1\\ 2\end{array}\right]$. Find v in ℝ³ such that $\left[\begin{array}{ccc}1& -3& 4\\ 2& -6& 8\end{array}\right]=u{v}^T$.