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No Similarity-Invariant Matrix Norm

September 4th, 2009

A matrix norm on Mn is said to be weakly unitarily-invariant if conjugating a matrix by a unitary U does not change the norm. That is,

\|X\|=\|UXU^*\|\ \ \forall \, X,U\in M_n \text{ with $U$ unitary.}

Many commonly-used matrix norms are weakly unitarily-invariant, including the operator norm, Frobenius norm, numerical radius, Ky Fan norms and Schatten p-norms. One might naturally wonder whether there are matrix norms that satisfy the slightly stronger property of similarity-invariance:

\|X\|=\|SXS^{-1}\|\ \ \forall\, X,Sin M_n\text{ with $S$ nonsingular.}

Upon first glance there doesn’t seem to be any reason why this shouldn’t be possible — one can look for simple examples that cause problems, but you’ll have trouble coming up with a matrix that causes problems if you restrict your attention to “nice” (i.e., normal) matrices. Nevertheless, we have the following lemma, which appeared as Exercise IV.4.1 in [1]:

Lemma (No Similarity-Invariant Norm). Let f : Mn → R be a function satisfying f(SXS-1) = f(X) for all X,S ∈ Mn with S invertible. Then f is not a norm.

If you’re interested in the (very short and elementary) proof of this lemma, see the pdf attached below. I would be greatly interested in seeing a proof of this fact that relies less on the structure of matrices themselves. It seems as though there should be a more general result that characterizes when we can and can not find a norm on a given vector space that is invariant with respect to some given subgroup, or some such thing. Would anyone care to enlighten me?

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References:

  1. R. Bhatia, Matrix analysis. Volume 169 of Graduate texts in mathematics (1997).
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