Is there a name for a matrix formed by multiplying a column and a row?

Question

I'm in the process of exploring Bra Ket notation. In it, I often find operators in the form $\lvert a\rangle\langle b\rvert$, which can be thought of as multiplying a row vector $a$ with a column vector $b$.

This strikes me as a construction which should probably have a name that I can research to understand the properties of matrices formed this way, but I'm having trouble finding sources that name such matrices.

What is it called when a matrix can be decomposed into a row vector and a column vector? I'd like to look up the properties of such a matrix.

$$M=\begin{pmatrix} a_0 \\ a_1 \\ \vdots \\ a_n \\ \end{pmatrix} \begin{pmatrix} b_0 & b_1 & \ldots & b_n \\ \end{pmatrix}$$

Answer 1

It can be shown that a matrix $M$ has rank equal to one if and only if $M = ab^\top$, where $a$ and $b$ are column vectors with complex entries, so the matrices you are thinking of can be referred to as rank-one matrices.

The product $ab^\top$ is also known as the outer product whereas the product $a^\top b$ is known as the inner product of $a$ and $b$.

Answer 2

The class of matrices are precisely the matrices with rank $1$, and the matrix $a b^{\mathsf T}$ specifically is called the outer product of $a$ and $b$ (by analogy with the inner product $a^{\mathsf T}b$, which gives a scalar).

Especially in the context of quantum states, it is also common to identify the vector space of $n \times n$ matrices with the tensor product $\mathbb R^n \otimes \mathbb R^n$, in which case the matrix $a b^{\mathsf T}$ corresponds to the pure tensor $a \otimes b$.