Orthogonal Matrices Explained: Examples & More

[captain]

what exactly are orthogonal matrices? can someone give me an example of how they would look like?

 

[matt grime]

An orthogonal matrix is one that satisfies XX^t = Id. You can generate examples yourself very easily.

 

[Chris Hillman]

The one theorem everyone should know concerning orthogonal matrices

Everyone knows that a surprising fact about three dimensional special orthogonal matrices is that they fix pointwise a one-dimensional subspace and act in the orthogonal two-dimensional subspace just like a two-dimensional rotation. But what is the n-dimensional generalization?

Consider a block diagonal matrix with one two by two block, which is an ordinary two-dimensional rotation matrix, plus ones down the diagonal. Call this a "rotation matrix"; it is a special case of a "special orthogonal matrix", i.e. an element of $SO(n)$. Geometrically, it fixes pointwise a codimension two subspace orthogonal to the two-plane, and acts like a two-dimensional rotation orthogonal to this "axis". It is possible to decompose a special orthogonal matrix as a product of "rotation matrices" all respecting a particular orthogonal direct sum decomposition of the vector space $R^n$ into two-dimensional subspaces (with a one-dimensional pointwise fixed subspace left over in case of odd dimension). In general, two distinct special orthogonal matrices will require two distinct orthogonal direct sum decompositions; this observation generalizes the fact that two elements of $SO(3)$ will usually have "rotation axes" pointing in different directions. However, the elements of $SO(n)$ which do share such a decomposition form an abelian subgroup. This in fact gives a large conjugacy class of abelian subgroups

An interesting example: apply this idea to the permutation matrix corresponding to an n-cycle in $R^n$.

Once this theorem is established, it is easy to see how to modify it to obtain a similar decomposition for any element of $O(n)$.

I just checked two dozen books which discuss the orthogonal group, including Birkhoff and Mac Lane, A Survey of Modern Algebra (chapter 9), Armstrong, Groups and Symmetry (chapters 9 and 19), Neumann, Stoy and Thompson, Groups and Geometry (chapters 14,15), Jacobson, Basic Algebra I (chapter 9), and Artin, Geometric Algebra (chapter III) and unfortunately was unable to find any mention of this. Wikipedia doesn't mention anything like this either. Yet it is a quite well known nineteenth century theorem. Go figure...

This is perhaps the most elementary thing one can say in discussing what elements of the orthogonal group $O(n)$ look like and how they act on $R^n$

(Edit: finally found a citation for you. The theorem is stated without proof in Senechal, Quasicrystals and Geometry, Prop 2.12, p. 47; see p. 63 for the decomposition of a five-cycle in $R^5$. Geometrically speaking, the effect of this element of $SO(5)$ respects an orthogonal direct sum decomposition into one pointwise fixed line plus two two-dimensional subspaces; it acts like a one-fifth turn in one of these, and like a two-fifth turn in the other. By linearity this description extends to all of $R^5$. Senechal cites P. Engel, Geometric Crystallography, Reidel, 1986.)

 

[mathwonk]

an orthogonal real matrix is one which defines a length preserving self transformation of R^n which fixes the origin, such as a composition of rotations and reflections about fixed sets containing the origin.\The theorem classifying them is one of the few things in herstein's topics in algebra that is not in most other books.

 

[captain]

if orthogonal matrices are for rotation them what unitary matrices for (or unitary groups)?

 

[mathwonk]

i don't know, since i have little intuition for complex transformations.

 

[Chris Hillman]

Topics in Algebra, just what I was looking for, plus a warning

The theorem classifying them is one of the few things in herstein's topics in algebra that is not in most other books.

Saw your other post, and I loved that book when I was an undergraduate! The textbook in question, Herstein, Topics in Algebra, does indeed cover not just Jordan form but also rational form, and... hrm... oh yes, I overlooked p. 348, which has the decomposition of an element of O(n). So there you go, that's the citation I was looking for. Thanks, mathwonk!

For the OP: in reading algebra textbooks, be very careful about left versus right actions. Zillions of unwary math students have fallen victim to this notational glitch, in part because very few authors even of textbooks bother to mention the issue!

To wit: Herstein uses right actions in discussing permutations in Topics, which means that you read composition left to right. This is just what you want if you use GAP (a symbolic computation package used by many for computational group theory, ring theory, and so on), or if you are "reading" a Cayley or Schreier graph (depicting via color-coded directed edges the effect of the generating elements in an action by a finitely generated group on some set). But if you are using "composition" (fog)(x) = f(g(x)), you must read composition right to left, which means you must use left actions. That's what Herstein does in his other textbook, Abstract Algebra, one of the very best "short" modern algebra textbooks.

(I bet mathwonk can think of a well-known theorem from covering space theory which uses both right and left actions; also, there is a simple trick for converting left to right actions or vice versa, but the notational issue is genuine and not so easily evaded.)

 

[Chris Hillman]

Unitary matrices: who needs 'em?

if orthogonal matrices are for rotation them what unitary matrices for (or unitary groups)?

Or as Pauli once asked, wrinkling his nose at the appearance of the nuetrino, "who ordered that?" A good question!

John Baez says he plans to devote future Weeks to discussing Kleinian geometry, particularly geometries related to the classical groups, which include the orthogonal, unitary, and symplectic groups. If so, no doubt he'll give a much better answer than I can hope to provide, but here's a "cheap shot":

The unitary group U(n) is a Lie group, i.e. a smooth manifold as well as a group. In fact it is a compact Lie group, so (by an important theorem in Lie theory/measure theory) it has a unique bi-invariant probability measure, Haar measure. (Bi-invariant means it respects left and right action by the group on itself by group multiplication--- speaking of left versus right, here's another place where this distinction is actually important, yet almost all authors ignore it as I said!) This is used to model the evolution over time of a quantum system having no particular symmetries. O(n) is a subgroup of U(n) and the coset space U(n)/O(n) inherits an invariant probability measure, which is used to model the evolution over time of a quantum system having time reversal symmetry.

For you computer programmers out there, at one time or another you probably have been required to generate random elements of the unitary group. And you've probably done it wrong. See Francesco Mezzadri, "How to Generate Random Matrices from the Classical Compact Groups", Notices of the AMS 54 (2007): 592--604, from which I got the above (clearly inadequate) physical interpretation.