# Can you write this operation in matrix notation?

Consider the matrix $$A$$ and column vector $$b$$ where,

$$A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$$ and $$b = \begin{bmatrix}1\\3\end{bmatrix}$$

In the R statistics software, the the code A*b performs element-wise multiplication of the columns in $$A$$ with the column vector $$b$$ returning a matrix with the same dimensions as $$A$$, i.e.

$$A*b = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} * \begin{bmatrix}1\\3\end{bmatrix} = \begin{bmatrix}1\times1 & 2\times1\\3\times3 & 4\times3\end{bmatrix}=\begin{bmatrix}1 & 2\\9 & 12\end{bmatrix}$$

Is there a name for this operation? Can this operation be expressed using common matrix notation?

## 2 Answers

It can be written as

$$A\circ(bu)$$ where $$u$$ is row vector of ones and $$\circ$$ is the hadamard product

Let us call $$\text{Diag}(b)$$ the diagonal matrix, the diagonal of which is $$b$$

$$\text{Diag} (b) = \begin{bmatrix} 1 & 0\\0 & 3\end{bmatrix}$$

Then

$$\text{Diag} (b) \times A = \begin{bmatrix} 1 & 2\\9 & 12\end{bmatrix}$$