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A Backward Triangle Inequality for Matrices

This month’s Lemma of the Month comes all the way from the Summer School and Advanced Workshop on Trends and Developments in Linear Algebra in Trieste, Italy, where Professor Rajendra Bhatia presented a lecture that introduced several simple yet endlessly interesting matrix inequalities. I will briefly present the various results here without proof, though the proof of the first “stepping stone” lemma is provided in the PDF attached to the bottom of this post. The truly interested reader can find full proofs in Professor Bhatia’s notes (follow the link above) or in [1].

Recall that one of the defining properties of a matrix norm is that it satisfies the triangle inequality:

Triangle Inequality

So what can we say about generalizing the backward triangle inequality to matrices? We can of course replace A by A – B in the above equation to find the following backward triangle inequality:

Backward Triangle Inequality

However, what happens if we swap the roles of the absolute value and the matrix norm on the left-hand side? That is, if we recall that |A| is the positive semidefinite part of A (i.e., the square root of A*A), then can we say something like

Incorrect Backward Triangle Inequality

It turns out that the answer to this question is heavily norm-dependent, so we will focus on the norm that gives the simplest answer: the Frobenius norm, which I will denote by ||•||2.

Theorem [Araki-Yamagami]. Let A, B ∈ Mn. Then

Araki

Finally, it just wouldn’t seem right to post from Italy without sharing a bit of it. Thus, I leave you with a taste of the highlight of the trip (excepting the mathematics, of course).

Building Up to the Result

In order to prove the result, one can proceed by proving a series of simple commutant-type matrix norm inequalities, which are interesting in their own right.

Lemma. Let A, B ∈ Mn be positive semi-definite and let X ∈ Mn be arbitrary. Then

Lemma 1

Lemma. Let A ∈ Mn be Hermitian. Then

Lemma 2

Lemma. Let A, B ∈ Mn be Hermitian. Then

Lemma 3

Finally, it just wouldn’t seem right to post from Italy without sharing a bit of it. Thus, I leave you with a taste of the highlight of the trip (excepting the mathematics, of course).

The colosseum as of seven hours ago

The colosseum as of eight hours ago

Related Links

References

  1. H. Araki and S. Yamagami, An inequality for the Hilbert-Schmidt norm, Commun. Math. Phys., 81 (1981) 89-98.
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