Outer product - Knowlpedia

For real column vectors $v\in\mathbb R^m$ and $w\in\mathbb R^n$ , their outer product is

v\otimes w=vw^T,\qquad (v\otimes w)_{ij}=v_iw_j.

It acts on a vector $x\in\mathbb R^n$ by $(v\otimes w)x=v(w\cdot x)$ . Its range is contained in the span of $v$ , and its rank is one when both vectors are nonzero.

Quadratic tensors

The matrix $v\otimes v$ is symmetric and positive semidefinite since $x^T(v\otimes v)x=(v\cdot x)^2\ge0$ . For complex vectors, a Hermitian covariance uses $vv^*$ , with conjugation, whereas the bilinear tensor $vv^T$ has a different meaning. State that convention explicitly.

References

Sheldon Axler, Linear Algebra Done Right, 4th ed., Chapters 2–3, 6 and 9.