Linear Algebra: Dimensions Vs. Degrees of Freedom
The last look we took at representation theory in the scope of this series arguably fell a bit short. The method for decomposition was good, but without all the basics that would be involved with that, that's not worth a lot. This month will go into the basics of linear algebra and matrices, maybe in a somewhat granular way, but that will help tremendously in making representation theory a powerful analytical tool. I shall not explain what a matrix is, and instead do a number of examples, so the reader can intuit.
The degrees of freedom of any mathematical object is the number of elements one can choose freely before the object is fully defined. This is not a property confined to matrices or similar, much rather it's a measure of the complexity of a problem. Setting a certain number of free variables will automatically restrict the options of the remaining variables, until a singular value is the only remaining option left. For example, the equation y = 2x + 4 has two unknown variables, but setting either one will automatically define the other for the statement to be true. This equation then has 1 degree of freedom. Other considerations of this type will work with constraints restricted onto the object. They are, in other words, a matter of definition. One could for example, define a vector in two dimensions which will have 2 degrees of freedom, unless we define y to be exclusively dependent on x, for example. This will eliminate one degree of freedom. Similar, if we know y to be a known constant (e.g. 0), then the vector will have one degree of freedom as well. Such constraints will of course become more complex as the object gains in complexity. Matrices for example gain properties like Traces, or symmetry, all of which will create constraints eliminating degrees of freedom. In a 3x3 matrix with no known elements, adding tracelessness for example will eliminate one degree of freedom, as tracelessness will require the diagonal entries to add up to 0. Requiring symmetry or antisymmetry will eliminate another degree of freedom, as knowing the upper right entry will automatically determine the lower left one.
Matrices can be written as essentially a system of equations. This means that there are internal operations within the matrix that don't actually change the space that they describe, but only its representations. At the same time this implies that any space will have several matrix representations. Within the system of equations, the operations that are allowed are equivalent to those allowed within the Gauss-algorithm. This is a fundamental analytical method and I won't get into it here, but keep in mind these operations for the following discussions.
The dimension of matrix space refers to the dimension of space contained in the matrix. As a system of equations, arranged in a way that each coefficient might as well be associated with a coordinate, a matrix could be conceived as a collection of vectors. To determine the dimension of that space then, one need only count the independent vectors. Vectors that are included in the matrix, but not independent can be reduced to a zero row by the Gauss operations. In a way, finding a zero-row is one of the most definitive ways to reduce the degrees of freedom, since that sets one of the dimensions completely to zero, removing them from further considerations.
The complementary property to this is the column rank of a matrix. Whereas the dimension is determined by the number of irreducible rows, the rank is determined by the number of irreducible columns. Because of the layout of coordinates within the matrix, the Gauss operations can be done along either axis. This is implies that the rank and dimension of matrix space are actually the same. It then makes sense to describe the dimension of the matrix space as "rank", and the approach in the preceding paragraph as "row rank". Note, that the space spanned by the row vectors and the column vectors don't need to be the same. In fact, their inequality is what qualifies matrices as transformation operators in the first place.
In larger dimensions, invertibility is no longer a trivial property. Several matrices don't actually have an Inverse in the space they're defined in. There are several ways of arriving at the inverse of a matrix, but I find the most versatile to be fueled by the Gauss algorithm again. Take any matrix and apply the identity on the right. Applying equivalent Gauss operations to get the identity on the left side, will slowly build the inverse of the matrix in question on the right. If this at any point proves impossible, there is no inverse. Sometimes however, one is really more interested in the existence of the inverse, rather than in its explicit definition. This will be the case in determining whether it's element of some group or other. If the inverse does not exist, then it definitely isn't. Note that here the identity property is implicit through the method, as just by writing it down, you would be finished. It is, insofar, a very illustrative method.
In the realm of square matrices, matrices have an extra property, which goes hand in hand with invertibility, this being the Determinant. Invertible matrices have a nonzero determinant, and are often very easy to compute. Ultimately, almost all ways of computing the determinant do essentially the same, but use increasingly obscure expressions to write them down. I'm not too great at keeping track of permutations, so I tend to stay away from the ε-tensor notation, and stick with the Leibniz formula.
What does it mean for a matrix to be uninvertible? Well, a matrix as a kind of operator will project one space onto another. It is, in a sense, a linear map. Having an inverse implies bijectivity, which qualifies the matrix as an isomorphism. This seems to check out when considering their applications as transformation operators, which are always defined as isomorphisms between spaces or representations. The determinant of a matrix is always a number, which, under careful consideration, is the area of the 2 ⨉ 2 matrix is equal the area of a parallelogram where the entries make up the parameters. The entries of the matrix will give the dimensions of the parallelogram, which could be understood as a skewing of the euclidean space. This can be generalized to all n ⨉ n matrices. Having a zero determinant, implies that the resulting surface has zero area, so it might not even live in the prescribed dimension.
Since matrices are very flexible in how they are displayed, there mathematicians have figured out forms that are more advantageous than the general case. One such case is that of matrix decomposition. Like any other decomposition, matrix decomposition is the practice of rewriting a matrix into a product of (convenient) matrices, which, if included in a product, enable some generalized statement about the end product. This is the actual mathematical basis for representation theory.
For a good grasp of representation theory, it's advantageous to know what a group's decomposition, and what being an element of that group entails. The first kind of matrices that should come to mind, is the unitary group of n ⨉ n matrices U(n). An element of U(n) is such, that it's complex conjugated transpose matrix is also its inverse. To confirm that this is a group, one only needs to realize that the inversion is a linear operation. A subgroup of the unitary group is the special uniyary group SU(n), which requires the determinant to be 1. For different dimensions, each of these groups (trivially) have a different number of generators.
There's a somewhat unintuitive way to express a unitary matrix through exponentiating a hermitian matrix. Hermitians are equal to their conjugate transpose, so they have n(n+1)/2 real and n(n-1)/2 imaginary components, so n^2 independent elements. These get handed down to the unitary matrices they create, although unitarity adds one more constraint, so the number of generators for SU(n) is n^2 - 1. The consideration concerning U(n) is over very quickly, seeing as they function basically identically to Hermitian matrices, although their conjugate transpose comes out to be its own negative. That minus sign need not bother us for this cause though.
Access to general terms for dimensions are given to us by representation theory, and when using representation theory, there is usually a silent implication we move on some kind of spacetime. That doesn't always have to tell us much, but we can be sure that the space has an irreducible representation as a metric, which we might as well use, seeing as the irreducible representations are equivalent. We have discussed this in the past. Let's start with the standard Minkowski metric. The matrix for the minkowski metric is defined very simply: A square diagonal matrix, notated as diag(-1, 1, 1..., 1). Most metrics are defined very narrowly. More often than not they have exactly one matrix to describe the space-time topology. Dimension and degrees of freedom are then locked to space dimensions + 1 and 0 respectively. However, things change when we place another object onto the metric backgroud. We stick to the Minkowski again and start by sticking a vector field on there. In general, the degrees of freedoms of any vector on a metric background is the same as the dimension of the metric. The degrees of freedom of any vector field on that metric background will then be at them most the dimension.
As a note, the rules for valid matrix multiplication is a great help when thinking about the dimensions and degrees of freedom. Because each matrix multiplication can be viewed as a coordinate transformation, coordinate transformations won't actually change the degree of freedom of some vector between representations. Something similar will happen for the change of metrics, though the change may be very much more jarring for the derivation aspect that is often non-trivial for useful metrics.