Htykuut

Suppose I am coding in python.

Then one can do e.g. something like this:

import numpy as np

A = np.random.rand((6,6))

# Lets reshape it

A_new = A.reshape(-1,3)

So it went from 2D array (matrix) to a 2D array with six 12 rows and three columns.

How would one write the re-shape operation formally in mathematical notation? Transpose is obviously easy, but is there a parallel for reshapes?

$$mathbf{A} in mathbb{R}^{6times6} rightarrow mathbf{A} in mathbb{R}^{12 times 3}$$

Thx

The operation $mathbb{R}^{ntimes m}tomathbb{R}^{nm}$ is called vectorization and often is denoted with $mathop{mathrm{vec}}()$. Creating a matrix from vector is an inverse $mathop{mathrm{vec}^{-1}}()$. When one can't deduce what are the dimensions of new matrix, they can be given as sub-indices: $mathop{mathrm{vec}^{-1}_{3,4}}(mathbf v)$ will produce the matrix $3times 4$. To sum up, you can write:
$$
mathop{mathrm{vec}^{-1}_{12,3}}circmathop{mathrm{vec}}: mathbb{R}^{6times 6}tomathbb{R}^{12,3},\
A'= mathop{mathrm{vec}^{-1}_{12,3}}(mathop{mathrm{vec}}(A)).
$$

However, I need to mention, that it isn't a widely used notation, so you are better to introduce it formally beforehand.

Note. There is also a term tensor reshaping, but it's usually referred to an operation of combining dimensions (so no dimension is needed to be a composite integer).