How to interpret symmetrical $2\times2$ matrix geometrically as a composition of simple transformation?

Question

Let's say I have this matrix:

$$A = \left[ \begin{matrix} 1&3\\ 3&1 \end{matrix} \right]$$

I only know simple transformation like reflection, rotation, scaling, and shear. Can I interpret matrix $A$ as a composition of these simple transformation? How?

I have tried to imagine the transformation of matrix $A$ and intuitively, it looks some kind of reflection or rotation with some scaling, but I got stuck because it has -2 and 4 scaling, which makes me confused.

Answer 1

All symmetric matrices (of any dimenstion, not just $2 \times 2$) can be thought of as a rotation followed by a scaling and/or reflection along each coordinate axes followed by the inverse of the first rotation.

In the case of your matrix, the sequence of transormations is

1. Rotate by $-\pi/4$
2. Scale in the $x$-axis direction by $4$ and in the $y$-axis direction by $-2$ (ie., scale by $2$ and then reflect)
3. Rotate back by $\pi/4$.

As a produce of simple matrices, this is \begin{align*} \begin{pmatrix} 1 & 3 \\ 3 & 1 \end{pmatrix} &= \begin{pmatrix} \cos(\pi/4) & -\sin(\pi/4) \\ \sin(\pi/4) & \cos(\pi/4) \end{pmatrix} \begin{pmatrix} 4 & 0 \\ 0 & -2 \end{pmatrix} \begin{pmatrix} \cos(-\pi/4) & -\sin(-\pi/4) \\ \sin(-\pi/4) & \cos(-\pi/4) \end{pmatrix} \\ &= \begin{pmatrix} 1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix} \begin{pmatrix} 4 & 0 \\ 0 & -2 \end{pmatrix} \begin{pmatrix} 1/\sqrt{2} & 1/\sqrt{2} \\ -1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}. \end{align*}

If you haven't seen it already, the writing of a symmetric matrix as this kind of product is a very common and important procedure known as finding its eigendecomposition, where the scaling factors are known as the eigenvalues of the matrix.

Answer 2

The previous answer is great, but I would also suggest to read up a bit on the Singular Value Decomposition (SVD) to answer your question in more generality, but also because SVD is very important in all kinds of applications right now, and understanding it provides intuition for tons of problems.