Why are skew-symmetric matrices of interest?

Question

I am currently following a course on nonlinear algebra (topics include varieties, elimination, linear spaces, grassmannians etc.). Especially in the exercises we work a lot with skew-symmetric matrices, however, I do not yet understand why they are of such importance.

So my question is: How do skew-symmetric matrices tie in with the topics mentioned above, and also, where else in mathematics would we be interested in them and why?

Answer 1

This is not the area of math you're interested in, but here's an example I might as well write down. In convex optimization we are interested in the canonical form problem $$ \text{minimize} \quad f(x) + g(Ax) $$ where $f$ and $g$ are closed convex proper functions and $A$ is a real $m \times n$ matrix. The optimization variable is $x \in \mathbb R^n$. This canonical form problem is the starting point for the Fenchel-Rockafellar approach to duality.

The KKT optimality conditions for this optimization problem can be written as $$ \tag{$\spadesuit$} 0 \in \begin{bmatrix} 0 & A^T \\ -A & 0 \end{bmatrix} \begin{bmatrix} x \\ z \end{bmatrix} + \begin{bmatrix} \partial f(x) \\ \partial g^*(z) \end{bmatrix}, $$ where $g^*$ is the convex conjugate of $g$ and $\partial f(x)$ is the subdifferential of $f$ at $x$ and $\partial g^*(z)$ is the subdifferential of $g^*$ at $z$. The notation $\begin{bmatrix} \partial f(x) \\ \partial g^*(z) \end{bmatrix}$ denotes the cartesian product $\partial f(x) \times \partial g^*(z)$.

The condition $(\spadesuit)$ is a great example of a "monotone inclusion problem", which is a type of problem that generalizes convex optimization problems. The subdifferential $\partial f$ is the motivating example of a "monotone operator", but the operator $$ \begin{bmatrix} x \\ z \end{bmatrix} \mapsto \begin{bmatrix} 0 & A^T \\ -A & 0 \end{bmatrix}\begin{bmatrix} x \\ z \end{bmatrix} $$ is a good example of a monotone operator which is not the subdifferential of a convex function.

Answer 2

I don't know how much background knowledge you have, maybe all of this is known to you and you are looking for something else, but it's the first thing to come to my mind. I've tried to phrase the same statement in different ways.

The Lie algebra of skew-symmetric matrices is the Lie algebra corresponding to the Lie group of orthogonal matrices. In other words, the space of skew matrices is the tangent space at the identity of the manifold of orthogonal matrices. The space of skew matrices can in some sense be thought of as the infinitesimal version of orthogonal transformations. I don't have time to get into why this is useful in any detail, but in many cases Lie algebras are significantly easier to handle and still give a great deal of information about the corresponding group.

A part of all this is the following observation. If $A$ is a skew matrix, then its exponential, $\exp(A)$, is an orthogonal matrix.

There are many books and lecture notes available on Lie groups and algebras. Some sources can be found in the answers to this question.

Answer 3

Natural numerical discretizations of odd-order derivatives are skew-symmetric. Thus skew-symmetric matrices are important in the study of numerical methods for hyperbolic PDEs. They are also related to the properties of the PDEs themselves, since an odd-order derivative can be viewed as an infinite-dimensional skew-symmetric operator.

Answer 4

This is too long to be a comment to Peter's answer, but I'd like to point out that his answer has a vast generalisation.

Vaguely, the following holds: Any non-degenerate symmetric or skew-symmetric or hermitian or anti-hermitian form on an $n$-dimensional space $V$ gives rise to the notion of an adjoint with respect to that form, $A \mapsto A^\ast$, on the space of transformations of that space $A \in End(V)$, which we can write as $n \times n$-matrices. (For the standard inner product on $\mathbb R^n$, this adjoint is just the matrix transpose.) Things of utmost interest in mathematics and physics are Lie groups consisting of those transformations which respect such a form, meaning their adjoint is their inverse, $G^\ast = G^{-1}$. (For the standard inner product, we're looking at the classical orthogonal group $O_n$ or $SO_n$.) Then there is an abstract machinery which shows, vaguely said, that the Lie algebra corresponding to that Lie group consists of those transformations which are "skew" with respect to that form, meaning their adjoint is their negative, $A^\ast = -A$. (For the standard inner product, here you get the skew-symmetric matrices, and that's what Peter's answer talks about.)

Now if one generalises this machinery far enough, there is a unifying theory of all the classical Lie groups and algebras (i.e. the types $A-D$, i.e. anything that eventually becomes something like $SL_n$ or $SO_n$ or $Sp_n$), and in this theory one has to switch back and forth between symmetric and skew-symmetric, hermitian and anti-hermitian, orthogonal and symplectic all the time. In a way, these notions just complement each other, and one gets a better view of everything when one allows oneself to go to the level of abstraction where they are variants of each other.

This theory is laid out best -- to my knowledge, I'm happy to learn of other sources -- in Knus, Merkurjev, Rost, Tignol: The Book of Involutions (if you're wondering "what involution?", it's the map $A \mapsto A^*$ above). I tried to give a very quick introduction to this theory in chapter 4.5 of my thesis.

Note: One thing that is easy to get confused about is that, by the crazy ubiquitous usefulness of matrices, they feature in this theory on at least two entirely different levels: 1) In the very special case of the standard inner product, $A^* = A^{T}$ (matrix transpose), the space of all skew-symmetric matrices form the Lie algebra $\mathfrak{so}_n$ to the special orthogonal group, see Peter's answer. But 2) also, each skew-symmetric matrix can be used to define a skew-symmetric form, and such a form defines one adjoint i.e. involution, and then one can look at the group of transformations which respect that form (which now might be a symplectic group $Sp_n$), and then its Lie algebra might be given by matrices which are "skew with respect to the skew form" (which might not be skew-symmetrical in the classical sense at all, rather resembling something orthogonal again).