Notation for sum of matrix with another matrix composed of copies of a vector

Question

I want to sum matrix $\mathbf{A}$ ($n \times d$) and a matrix composed of $n$ copies of vector $\mathbf{b}$ ($d \times 1$) as follows:

$\mathbf{A}+\begin{bmatrix}\mathbf{b^T \\ \vdots \\ b^T}\end{bmatrix}_{n \times d}$

In MATLAB or Julia this would be equivalent to:

A .+ b'

Is there any succinct mathematical notation for this operation, or is the notation I used here the most appropriate?

Thanks in advance.

mjw · Accepted Answer · 2021-08-17 19:46:01Z

1

Let $\mathbf{1}= \pmatrix{1\\1\\ \vdots \\1}$

You want $\mathbf{A}+ \mathbf{1} \, \mathbf{b}^T$

answered Aug 17, 2021 at 19:46

8,8611 gold badge9 silver badges28 bronze badges

mathcounterexamples.net · Accepted Answer · 2021-08-17 19:46:07Z

If $\mathbf{1}$ denotes the $n \times 1$ vector with all entries equal to $1$, you could write

$$\mathbf{A}+\begin{bmatrix}\mathbf{b^T \\ \vdots \\ b^T}\end{bmatrix}_{n \times d} = A + \mathbf{1} \otimes \mathbf{b} $$ using tensor product.

2 Answers 2