Balancing circles

After extensive testing with glasses of water, I’ve determined that it’s impossible to stably balance one infinitely thin circle on top of another. This is true whether the circles are the same size or different, and does not depend on the mass distribution of the circles. This is because two circles intersect at either zero or two points (unless they are the same, and that still isn’t a stable configuration).

Matrix plus scalar

If $a$ is a scalar and $M$ is a square matrix, it is very convenient to be able to write $a + M$. Usually people know immediately what this means, but are uneasy about “abusing” notation, so here’s the detailed justification for why this is perfectly legitimate: Matrices should be considered first and foremost as linear transformations. You know what a matrix is if you know what it does to vectors.

Singular values are not the magnitudes of eigenvalues

A week ago someone asked whether the singular values of a general (real) matrix are the magnitudes of its eigenvalues. There are various ways to see that the answer is no, but here’s an amusingly nonconstructive proof: Consider a random matrix $A$ taken from $GL(n)$ with some smooth distribution. With probability 1 all singular values of $A$ will be unique. However, with nonzero probability $A$ will be near a rotation matrix and will have a complex conjugate pair of eigenvalues with the same magnitude.