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Isometries of Unitarily-Invariant Complex Matrix Norms

August 15th, 2010

Recall that a unitarily-invariant matrix norm is a norm on matrices X ∈ Mn such that

One nice way to think about unitarily-invariant norms is that they are the matrix norms that depend only on the matrix’s singular values. Some unitarily-invariant norms that are particularly well-known are the operator (spectral) norm, trace norm, Frobenius (Hilbert-Schmidt) norm, Ky Fan norms, and Schatten p-norms (in fact, I would say that the induced p-norms for p ≠ 2 are the only really common matrix norms that aren’t unitarily-invariant – I will consider these norms in the future).

The core question that I am going to consider today is what linear maps preserve singular values and unitarily-invariant matrix norms. Clearly multiplication on the left and right by unitary matrices preserve such norms (by definition). However, the matrix transpose also preserves singular values and all unitarily-invariant norms – are there other linear maps on complex matrices that preserve these norms? For a more thorough treatment of this question, the interested reader is directed to [1,2].

Linear Maps That Preserve Singular Values

We first consider the simplest of the above questions: what linear maps Φ : Mn → Mn are such that the singular values of Φ(X) are the same as the singular values of X for all X ∈ Mn? In order to answer this question, recall Theorem 1 from my previous post, which states [3] that if Φ is an invertible map such that Φ(X) is nonsingular whenever X is nonsingular, then there exist M, N ∈ Mn with det(MN) ≠ 0 such that

In order to make use of this result, we will first have to show that any singular-value-preserving map is invertible and sends nonsingular matrices to nonsingular matrices. To this end, notice (recall?) that the operator norm of a matrix is equal to its largest singular value. Thus, any map that preserves singular values must be an isometry of the operator norm, and thus must be invertible (since all isometries are easily seen to be invertible).

Furthermore,  if we use the singular value decomposition to write X = USV for some unitaries U, V ∈ Mn and a diagonal matrix of singular values S ∈ Mn, then det(X) = det(USV) = det(U)det(S)det(V) = det(UV)det(S). Because UV is unitary, we know that |det(UV)| = 1, so we have |det(X)| = |det(S)| = det(S); that is, the product of the singular values of X equals the absolute value of its determinant. So any map that preserves singular values also preserves the absolute value of the matrix determinant. But any map that preserves the absolute value of determinants must preserve the set of nonsingular matrices because X is nonsingular if and only if det(X) ≠ 0. It follows from the above result about invertibility-preserving maps that if Φ preserves singular values then there exist M, N ∈ Mn with det(MN) ≠ 0 such that either Φ(X) = MXN or Φ(X) = MXTN.

We will now prove that M and N must each in fact be unitary. To this end, pick any unit vector x ∈ Cn and let c denote the Euclidean length of Mx:

By the fact that Φ must preserve singular values (and hence the operator norm) we have that if y ∈ Cn is any other unit vector, then

Because y was an arbitrary unit vector, we have that N* = (1/c)U, where U ∈ Mn is some unitary matrix. It can now be similarly argued that M = cV for some unitary matrix V ∈ Mn. By simply adjusting constants, we have proved the following:

Theorem 1. Let Φ : Mn → Mn be a linear map. Then the singular values of Φ(X) equal the singular values of X for all X ∈ Mn if and only if there exist unitary matrices U, V ∈ Mn such that

Isometries of the Frobenius Norm

We now consider the problem of characterizing isometries of the Frobenius norm defined for X ∈ Mn by

That is, we want to describe the maps Φ that preserve the Frobenius norm. It is clear that the Frobenius norm of X is just the Euclidean norm of vec(X), the vectorization of X. Thus we know immediately from the standard isomorphism that sends operators to bipartite vectors and super operators to bipartite operators that Φ preserves the Frobenius norm if and only if there exist families of operators {Ai}, {Bi} such that Σi Ai ⊗ Bi is a unitary matrix and

It is clear that any map of the form described by Theorem 1 above can be written in this form, but there are also many other maps of this type that are not of the form described by Theorem 1. In the next section we will see that the Frobenius norm is essentially the only unitarily-invariant complex matrix norm containing isometries that are not of the form described by Theorem 1.

Isometries of Other Unitarily-Invariant Norms

One way of thinking about Theorem 1 is as providing a canonical form for any map Φ that preserves all unitarily-invariant norms. However, in many cases it is enough that Φ preserves a single unitarily-invariant norm for it to be of that form. For example, it was shown by Schur in 1925 [4] that if Φ preserves the operator norm then it must be of the form described by Theorem 1. The same result was proved for the trace norm by Russo in 1969 [5]. Li and Tsing extended the same result to the remaining Schatten p-norms, Ky Fan norms, and (p,k)-norms in 1988 [6].

In fact, the following result, which completely characterizes isometries of all unitarily-invariant complex matrix norms other than the Frobenius norm, was obtained in [7]:

Theorem 2. Let Φ : Mn → Mn be a linear map. Then Φ preserves a given unitarily-invariant norm that is not a multiple of the Frobenius norm if and only if there exist unitary matrices U, V ∈ Mn such that

References:

  1. C.-K. Li and S. Pierce, Linear preserver problems. The American Mathematical Monthly 108, 591–605 (2001).
  2. C.-K. Li, Some aspects of the theory of norms. Linear Algebra and its Applications 212213, 71–100 (1994).
  3. J. Dieudonne, Sur une generalisation du groupe orthogonal a quatre variables. Arch. Math. 1, 282–287 (1949).
  4. I. Schur, Einige bemerkungen zur determinanten theorie. Sitzungsber. Preuss. Akad. Wiss. Berlin 25, 454–463 (1925).
  5. B. Russo, Trace preserving mappings of matrix algebra. Duke Math. J. 36, 297–300 (1969).
  6. C.-K. Li and N.-K. Tsing, Some isometries of rectangular complex matrices. Linear and Multilinear Algebra 23, 47–53 (1988).
  7. C.-K. Li and N.-K. Tsing, Linear operators preserving unitarily invariant norms of matrices. Linear and Multilinear Algebra 26, 119–132 (1990).
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