In matrix analysis, there are several different matrix norms that you might use depending on the context of your particular problem. If you are treating the matrix as an operator acting on a the complex vector space Cn, then you would likely use the operator norm. If you are considering the matrix as a density operator (i.e., if you’re a quantum information nerd like me) then you might want to use the trace norm. If you just want something that’s easy to calculate, you might be better off going with the Frobenius norm. These are three of the most well-studied and well-used matrix norms, and they have one very important thing in common — they are unitarily invariant. That is, if X ∈ M n , then

Unitarily-invariant norms are particularly “nice” in that they satisfy submultiplicativity as well as various other desirable properties. Here I will present two particular families of unitarily-invariant norms, briefly discuss some of their applications, and then define a family of norms that encompass all of the other norms mentioned in this post as special cases.

Before proceeding, recall that for any matrix X ∈ M n we can define the absolute value |X| of X to be the positive matrix square root of X*X. Then the singular values of X, s 1 (X), s 2 (X), …, s n (X), are defined to be the eigenvalues of |X|. Throughout this post we will assume that the singular values are ordered from largest to smallest (this is pretty standard practice when dealing with singular values):

Ky Fan Norms

Given a natural number k such that 1 ≤ k ≤ n, the Ky Fan k-norm of a matrix X ∈ M n is defined to be the sum of the k largest singular values of X:

While Ky Fan norms aren’t extremely well-known, they have applications is matrix theory as well as quantum information theory. For example, they have recently appeared in [1] as a tool for determining whether a linear map from M n to M m is k-positive, which is one of the difficult open problems in quantum information. If P k ⊆ M n denotes the space of rank-k orthogonal projections (i.e., matrices such that P2 = P* = P), then it is not difficult to show that

Several properties of these norms are obvious from the definition — for example, the Ky Fan k-norm is upper-bounded by the Ky Fan (k+1)-norm and each Ky Fan norm is unitarily-invariant. One property that isn’t immediately obvious, however, is the following very cool result:

Fan Dominance Theorem [2, Section IV.2]. Let X, Y ∈ M n . Then

if and only if

Schatten Norms

Given a real number p ≥ 1, the Schatten p-norm of a matrix X ∈ M n is defined to be the standard vector p-norm of the vector of singular values of X:

There are numerous applications of Schatten norms in quantum information theory. For example, they are used to define completely bounded norms for linear maps acting on matrices, which are probably the most important norms for maps in quantum information (see [3] for a particular paper that deals with these norms). As with the Ky Fan norms, the Schatten norms are unitarily-invariant and can be equivalently defined via an expression involving the trace:

One of the other nice properties of the Schatten p-norms is a modified submultiplicativity result, which states that if X,Y ∈ M n then

Everything In Between

We have now seen two families of norms based on the singular values of a matrix, both of which are very important in matrix analysis as well as quantum information theory. The Ky Fan norms are given by summing the first k singular values, while the Schatten norms are given by computing the standard vector p-norm of the vector of singular values. So why have I never seen the natural generalization of these two families of norms – the vector p-norm of the first k singular values – defined? (Update [May 14, 2012]: See the comments for a few references that study these norms.)

Definition. Let X ∈ M n , p ≥ 1 and 1 ≤ k ≤ n, with k a natural number. Then I define the (p,k)-singular norm of X to be

Notice that these norms are also unitarily-invariant, and as with the previously-defined norms, they are given by a relatively simple trace expression:

One particular case of these norms – the p = 2 case – actually appeared implicitly in [1], though they were referred to as Ky Fan norms. I have also found a need for the p = 2 case of these norms in a recent project of mine that will hopefully be wrapped up in the next month or so.

I will finish by pointing out some special cases of this norm:

If we allow p = ∞ by taking the limit as p → ∞ in the above definition, then the (∞,k)-singular norm coincides with the standard operator norm, regardless of k.

When p = 1, the (1,k)-singular norm is exactly the Ky Fan k-norm.

When k = n, the (p,n)-singular norm is exactly the Schatten p-norm.

When p = 1, k = n (i.e., the Schatten 1-norm, which equals the Ky Fan n-norm), we recover exactly the trace norm.

When p = 2, k = n (i.e., the Schatten 2-norm), we recover exactly the Frobenius norm.

When p = 1, k = 1 (i.e., the Ky Fan 1-norm), we again obtain the operator norm.

References