Notes

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. A scalar is also a linear transform on vectors: multiplying a scalar times a vector is a linear operation. Therefore, scalars can also be thought of as linear transformations, and therefore as matrices. It is immediate which matrix the scalar should be: the result of multiplying by a scalar $k$ is that all components are scaled by $k$; the matrix that does this is just $\mathrm{kI}$.

Here’s another way of saying that. Let $A$ be an associative algebra with unity over a field $K$. We can define a homomorphism of algebras $f:K\to A$ via $f(k)=k{1}_A$. $f$ is an isomorphism since if $j\ne k$, $f(j)=f(k)$ implies $0=(j-k{)}^{-1}(j-k)1_A=1_A$, a contradiction. $f$ is also the unique nontrivial homomorphism from $K$ to $A$, since algebra homomorphisms must send $1$ to either $0$ or $1$. Therefore, there is a unique copy of $K$ inside $A$, and we can identify $K$ with that copy.

This is a general principle: no one writes $(a+0i)+z$ when $a$ is real and $z$ is complex, because the complex numbers are considered a superset of the reals. The same is true of matrices: once we identify scalars with scalar diagonal matrices, matrices are just another superset of the reals.

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 $\mathrm{GL}(n)$ 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. Therefore, the singular values of $A$ are not always the magnitudes of the eigenvalues.

Basic math support appears to be working, so now I have a blog.

Here’s a test of inline math ($x+y+x_y+x^y$) and an integral:

Success!  I’ll provide details of the mathml/itex setup in a separate post.