Applications of non-square matrices

Question

I am wondering if non-square matrices have many applications.

It seems in my algebra classes we tend not to use them.

Answer 1

Yes; they're used all the time.

An example application: rows are labelled by stores, columns are labeled by items. Entries are the price of the item in the store.

If you have a (column) vector $v$ of quantities, and compute $Mv$, you get a (column) vector of the cost for your shopping at each store.

Light sources and surface patches in a scene in computer graphics: the $ij$ entry of $M$ says how much of the light from source $j$ gets to surface patch $i$. If you multiply by a column vector of source-brightnesses, you get out a column vector of patch brightnesses. (It's true that this is a crude approximation of real light transport, but something similar works in more generality).

Math application: you have an embedding of a surface in 4-dimensional space. The derivative of this map, at each point of the surface, can be represented by a $4 \times 2$ matrix.

Answer 2

Yes, in fact I would say that square matrices are quite rare!

For instance a map $\mathbb R^3\to \mathbb R^2$ projecting 3d-space onto a 2d-plane can be given as

$$\left(\begin{array}{ccc}1&0&0\\ 0 & 1&0\end{array}\right)$$ by which the vector $(a,b,c)$ is mapped to $(a,b)$. Second example: the scalar product of two vectors can be expressed conveniently as a matrix multiplication by the transpose:

$$\left(\begin{array}{c}a\\ b\end{array}\right)\cdot \left(\begin{array}{c}c\\ d\end{array}\right)=\big(a\quad b\big)\left(\begin{array}{c}c\\ d\end{array}\right)=ac+bd$$

There are many more examples!