Visual representation of matrices

There is no one way to "visualize" matrices. In fact, visualizing abstract quantities can be useful but also lead to loss of information through the visualization.

As far as 2x2 matrices go, one way of visualizing them is to note that they have the same mathematical structure as complex numbers when it comes to addition and multiplication (See http://en.wikipedia.org/wiki/2_%C3%97_2_real_matrices#2.E2.80.89.C3.97.E2.80.892_real_matrices_as_complex_numbers)

Of course, this is highly specific as far as matrices go. A more useful approach to "visualizing" a matrix is to view it a linear transformation on a vector -- and observe its action on a standard set of vectors. Linear transformations and their properties in terms of dilating and rotating vectors are useful ways to understand matrices, although they are not the only useful information contained inside a matrix.

I would recommend Artin's text as a useful first resource to highlight some of these connections. Depending on what you want to do with matrices, you can expand in many different directions.

You can represent a $2 \times 2$ matrices $A = \left[\begin{smallmatrix}a & b\\c & d\end{smallmatrix}\right]$ as a parallelogram $Q_A \subset \mathbb{R}^2$ with vertices $(0,0), (a,c), (b,d)$ and $(a+b,c+d)$. If one identify the plane $\mathbb{R}^2$ with $M^{2\times 1}(\mathbb{R})$, the space of $2 \times 1$ column matrices, then $Q_A$ is the image of the unit square $[0,1] \times [0,1]$ under linear transform

$$[0,1] \times [0,1] \ni \begin{bmatrix}x \\ y\end{bmatrix} \quad \mapsto \quad \begin{bmatrix}x' \\ y'\end{bmatrix} = \begin{bmatrix}a & b\\c & d\end{bmatrix} \begin{bmatrix}x \\ y\end{bmatrix} \in Q_A $$

Since a linear transformation is uniquely determined by its action on a basis, this provides a faithful represent of the $2 \times 2$ matrices. Under this representation, some geometry related operations now correspond to familiar geometric shapes. e.g.

• The matrix $\left[\begin{smallmatrix}s & 0\\0 & s\end{smallmatrix}\right]$ represents a scaling of geometric objects. It corresponds to a square of side length $s$, axis aligned with the standard $x$- and $y$-axis.

• The matrix $\left[\begin{smallmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \;\cos\theta\end{smallmatrix}\right]$ represents a counterclockwise rotation of angle $\theta$. It corresponds to the unit square rotated counterclockwisely for angle $\theta$.

• The matrices $\left[\begin{smallmatrix}1 & m\\ 0& 1\end{smallmatrix}\right]$ and $\left[\begin{smallmatrix}1 & 0\\ m& 1\end{smallmatrix}\right]$ represent sheer mappings in horizontal and vertical directions. They can visualized as parallelogram with one pair of its sides staying horizontal and vertical respectively.

This sort of shapes provides a useful visual mnemonics for what are the effects of those matrices (when viewed as a transformation of the plane).

Finally, one can also use this to introduce the concept of determinant to students.

• What is the the determinant of a matrix $A$? It is just the area of $Q_A$.
• What does it mean $\det A < 0$? It just mean the $Q_A$ has been flipped.

I'm sure there are other ways to use this visualization as a teaching tool.