Chapter 4.6, Problem 31E

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.

31. Verify that rank uv^T ≤ 1 if $u=\left[\begin{array}{c}2\\ -3\\ 5\end{array}\right]$ and $v=\left[\begin{array}{c}a\\ b\\ c\end{array}\right]$.