Row-Vector Matrix Product

linear-algebra

Definition

Row-Vector Matrix Product

Let $x\in F^{1\times m}$ be a row vector and $A\in F^{m\times n}$ a matrix over a field $F$. The row-vector matrix product is the operation

$$\cdot :F^{1\times m}\times F^{m\times n}\to F^{1\times n}$$

defined by

$$(x^{\top }A{)}_j=\sum _{i=1}^m{x}_i{A}_{ij}$$

for all $1\le j\le n$.

Einstein Notation: $(x^{\top }A{)}_j=x_i{A}_{ij}\in F^{1\times n}$