## Title

The title of this blog comes from a Japanese phrase often applied to the martial arts, but also to other disciplines such as the tea ceremony.

の — “no”, possessive suffix

Thus, 動中の静 means “the stillness of/in the midst of motion”, or “calmness in action”. It’s very difficult for me to articulate it precisely, but this seems to me to be among the cardinal qualities (virtues?) of the martial arts. The moments when I feel it in Aikido, Iaido or Zazen practice — or just in ordinary daily life — are sublime and priceless.

## Singular Value Decomposition

The singular value decomposition (SVD) of an m×n complex matrix A is a very useful decomposition of A into three factors. In some ways, it can be thought of as an extension of the ideas of eigenvalues and eigenvectors to non-square matrices. Recall that for A ∈ ℂn×n, λ ∈ ℂ and 0 ≠ x ∈ ℂn are an eigenvalue-eigenvector pair for A if Ax = λx. If, in addition, A is a normal matrix (i.e. AA = AA), then there is a unitary matrix V ∈ ℂn×n and a diagonal matrix Λ ∈ ℂn×n such that A = VΛV; perhaps unsurprisingly, the (normalized) eigenvectors of A are the columns of V, and the corresponding eigenvalues are the diagonal entries of Λ.

Can we do anything similar in the case that A is a non-normal square matrix, or even a non-square matrix? Remember that a matrix A ∈ ℂm×n represents a linear map from ℂn to ℂm with respect to fixed bases of either space. The idea behind the SVD is to separate the roles of the two spaces: we seek scalars σ ∈ ℂ (although they will later turn out to be non-negative reals) and non-zero vectors v ∈ ℂn and u ∈ ℂm such that

Av = σu and Au = σv

The scalar σ is called a singular value and u and v are called singular vectors; in some applications, these are more useful than eigenvalues and eigenvectors; note that the above equation is the eigenvalue problem if we insist that u = v ∈ ℂn. The idea now is to collect together all the singular values and vectors into appropriate matrices, just as in the eigenvalue/eigenvector case.

## MA398: Matrix Analysis and Algorithms

This autumn I will lecture the course MA398 Matrix Analysis and Algorithms at the University of Warwick. The course will be an introduction to matrix analysis and to the basic algorithms of numerical linear algebra, broadly following a text by Andrew Stuart and Jochen Voss [2009 version] [2012 version]. I plan on posting summary notes to this blog, roughly in correspondence with the chapters of the Stuart & Voss text; I’ll tag them all “ma398”.

After some necessary introductory material, the course will cover three basic problems of numerical linear algebra:

1. Simultaneous Linear Equations (SLE). Given a matrix A ∈ ℂn×n and a vector b ∈ ℂm, find x ∈ ℂn such that Ax = b.
2. (Overdetermined) Least Squares (LSQ). Given a matrix A ∈ ℂm×n with m ≥ n and a vector b ∈ ℂm, find x ∈ ℂn that minimises ‖ Ax − b ‖22.
3. Eigenvalue/Eigenvector Problem (EVP). Given a matrix A ∈ ℂn×n, find (xλ) ∈ ℂn×ℂ such that Ax = λx and ‖x2 = 1.

## Conic Sections

Recently I had a conversation about that classical area of geometry, the study of conic sections: ellipses, parabolae, and hyperbolae. Conics are pretty much the simplest plane curves that one can imagine after straight lines, and have some lovely connections to physical motions, the most well-known of which is that planets follow elliptical orbits. This post will quickly cover some of the basics.

Simply put, a conic section is the intersection of the double cone

K := { (x, y, z) ∈ ℝ3 | x2 + y2 = z2 }

with a (two-dimensional) plane P. The plane P is completely described by two things: its unit normal ν = (a, b, c) ∈ ℝ3 (“unit” because we insist that a2 + b2 + c2 = 1) and a signed distance d ∈ ℝ from the origin in ℝ3 to P along the direction ν. In this notation,

P = { (x, y, z) ∈ ℝ3 | ax + by + cz = d }.

So, we’re interested in C := KP, where K is the standard cone defined above and P is a plane in ℝ3. For the most part, this intersection is a one-dimensional smooth curve in P; there are a few degenerate cases that will get pointed out along the way. As it turns out, while the normal ν has a great influence on the shape of the curves C, the distance d is relatively unimportant: you get the same curve C simply scaled up or down if you make d larger or smaller; the only thing to watch out for is that things do change radically at d = 0. (Also, I’m restricting attention to the standard cone; if I were being really complete, then I’d consider the possibility that the cone might degenerate into a cylinder.)

## Heat

Heat (1995), written and directed by Michael Mann, is one of my all-time favourite films. I remember the circumstances under which I first saw it: in a hotel room in Florida in 1996, too young to properly appreciate the skill with which Mann was telling the story, but captivated by the end result all the same,

On the surface, Heat is a simple hunt/pursuit story, the tale of Los Angeles cop Vincent Hanna (played by Al Pacino) and professional thief Neil McCauley (Robert De Niro). Both are driven men who live for their chosen line of work:

“I do what I do best: I take down scores. You do what you do best: try to stop guys like me.” — Neil McCauley to Vincent Hanna

All the “standard” action movie elements are there: several robberies and shoot-outs, a car chase, a tense personal encounter between the two leads, vignettes illustrating the personal costs of their decisions on those around them, and of course a final and deadly one-on-one confrontation. The story itself is not terribly special or innovative, but Heat is a work of genius because of how well the story is told: it’s as much a classical tragedy in a modern setting as it is an action film.

## Downtime

I’m off the mat for a while, hopefully just a couple of weeks, while I rest an injury to my right knee. I’m no stranger to knee injuries, and in fact had this particular one in my left knee 5–6 years ago; that one eventually required surgical intervention, and I’m hoping that I’ve caught this one early enough to prevent the need for such drastic measures. Even so, having to take time off to heal is not exactly fun.