Geometric Interpretation of Matrix Additiom

Question

Is there a geometric meaning to matrix addition similar to how matrix multiplication acts as a linear transformation? I'm really curious thanks!

Answer 1

As far as I am aware, there's no clear geometric meaning attached to matrix addition other than in the specific cases. By that, I am referring to the addition of $n \times 1$ real matrices (which are basically just column vectors). That has a standard geometric interpretation in $\mathbb{R}^3$ and $\mathbb{R}^2$.

Now, you mentioned that 'matrix multiplication acts as a linear transformation'. That, in fact, is not very good mathematical wording, from what I can see

The idea here is that you begin with linear maps between vector spaces. There is a clear definition as to what constitutes a linear map. Then, you show that each linear map has an associated matrix that, when acting on a vector from the domain of the linear map, does exactly the same thing as the linear map does.

So, in a sense, a linear map is the same thing as its associated matrix and vice versa. You can talk about one by talking about the other and this is highly useful in proofs about matrices.

Matrix multiplication is, then, just seen as the composition of linear maps. In the same way, matrix addition is seen as the addition of two linear maps. In fact, you can prove many of the rules of matrix calculus by showing that they are consequences of the way maps combine.

For example, the composition of linear maps is not commutative. As a result, matrix multiplication isn't commutative.

There's a very beautiful way of looking at all of this using the language of sets & maps. I encourage you to look more into this. A good text that does go through it in a lot of detail is Klaus Janich's Linear Algebra.

Answer 2

As you note, each matrix corresponds to a linear transformation. Given a matrix $A$, the linear transformation that $A$ corresponds to is defined via matrix-vector mulitplication. In particular, if $A$ has size $m \times n$, then the corresponding transformation $T:\Bbb R^n \to \Bbb R^m$ is given by $T_A(x) = Ax$ for vectors $x \in \Bbb R^n$.

With that established, we can interpret the addition and multiplication of two matrices in terms of these linear transformations. In particular: if $A,B$ are $m \times n$ matrices, then the transformation $T_{A + B}$ corresponding to the matrix $A + B$ is defined so that $T_{A + B}(x) = T_A(x) + T_B(x)$. That is, $T_{A+B} = T_A + T_B$. If $A$ is $p \times m$ and $B$ is $m \times n$, then the transformation $T_{AB}(x) = T_A(T_B(x))$. That is, $T_{AB} = T_A \circ T_B$.

Since we're focusing on addition, here is an example: take $$ A = \pmatrix{1&0\\0&0}, \quad B = \pmatrix{0&0\\0&1}, \quad A+B = I = \pmatrix{1&0\\0&1}. $$ Note that $I$ is a special matrix known as the "identity matrix".

The transformations corresponding to $A$ and $B$ are as follows: $$ T_A(x) = \pmatrix{1&0\\0&0} \pmatrix{x_1\\x_2} = \pmatrix{x_1\\0}, \quad T_B(x) = \pmatrix{0&0\\0&1}\pmatrix{x_1\\x_2} = \pmatrix{0\\x_2}. $$ In other words: $T_A$ is the projection onto the $x_1$-axis and $T_B$ is the projection onto the $x_2$-axis. Adding these two transformations together means that we get the output for $x$ by adding $T_A(x)$ and $T_B(x)$. That is, we should have $$ T_{A+B}(x) = T_A(x) + T_B(x) = \pmatrix{x_1\\0} + \pmatrix{0\\x_2} = \pmatrix{x_1\\x_2}. $$ That is, adding $T_A$ and $T_B$ results in a transformation that produces the same input that we started with. This indeed matches up with the transformation corresponding to $A+B$. In particular, $$ T_{A+B}(x) = (A+B)x = \pmatrix{1&0\\0&1}\pmatrix{x_1\\x_2} = \pmatrix{x_1\\x_2}. $$