What is the geometric interpretation of matrix addition?

Question

I was studying linear algebra and trying to get a visual "feel" for it through watching 3Blue1Brown's "Essence Of Linear Algebra" series here

Essence Of Linear Algebra

Here, matrix multiplication is shown as the composition of 2 linear transformations. But I was wondering how we would interpret matrix addition/subtraction visually. I am not asking for $2$ by $1$ matrices since they would be equivalent to vector addition and subtraction respectively. But how would we interpret the addition of, say $2$ by $2$ matrices visually?

Answer 1

Here's an idea for square matrices:

Let $A$ and $B$ be two matrices, let $v$ be a 'compatible' vector i.e: one which is of the right size to be multiplied by A/B, then,

$$ (A+B)v = Av + Bv$$

The above you can think of in 'words', take a vector 'v', transform it according to some rule given by the matrices A, and similarly do it with B, then add these two results. Hence, we can understand the addition matrices by the action of the 'summed' matrix on a vector.

To contrast, matrix multiplication is just a rule for 'transforming vectors', basically take in a vector and convert it into a new one.

Answer 2

Consider, for example, that we are studying the soil of a rectangular area. Suppose there are 2 different layers of soil: a top layer that is near the surface, and another one that is underneath the top layer. The depth of each of these layers is different from one place to another.

We can divide the rectangular study area into a grid of cells, and identify each cell with its row and column in the grid. This way our study area will be like a matrix, with rows and columns and cells (elements). Then we can write our measurements in matrix form. For example, in the area grid, we go to the cell on row 1 and column 1, and measure the depth of the top layer of soil, and we write our measurement in element [1,1] of the matrix for the top layer. If we do this for all cells, we have a matrix summarizing the depth of the top layer of soil in all cells in the grid. In a similar way we can make a matrix summarizing the depth of the bottom layer of soil in all cells in the grid.

Now, if we add the two matrices, we are summing up the depths of the two layers of soil in each cell of the grid.