Ostrowski’s Theorem

[Bernice Ebesugawa]

Innumber theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q \displaystyle \mathbb Q is equivalent to either the usual real absolute value or a p-adic absolute value.[1]

(Note: In general, if x \displaystyle is an absolute value, x λ ^\lambda is not necessarily an absolute value anymore; however if two absolute values are equivalent, then each is a positive power of the other.[2]) The trivial absolute value on any field K is defined to be

For a prime number p, the p-adic absolute value on Q \displaystyle \mathbb Q is defined as follows: any non-zero rational x can be written uniquely as x = p n a b \displaystyle x=p^n\tfrac ab , where a and b are coprime integers not divisible by p, and n is an integer; so we define

where v p ( n ) \displaystyle v_p(n) is the p-adic valuation of n. The multiplicativity property enables one to compute the absolute value of n from that of the prime numbers using the following relationship

Another theorem states that any field, complete with respect to an Archimedean absolute value, is (algebraically and topologically) isomorphic to either the real numbers or the complex numbers. This is sometimes also referred to as Ostrowski's theorem.[3]

The eigenvalues of a Hermitian matrix are real and we order them . Note that in some references, such as Horn and Johnson (2013), the reverse ordering is used, with the largest eigenvalue. When it is necessary to specify what matrix is an eigenvalue of we write : the th largest eigenvalue of . All the following results also hold for symmetric matrices over .

with equality when is an eigenvector corresponding to and , respectively, This characterization of the extremal eigenvalues of as the extrema of is due to Lord Rayleigh (John William Strutt), and is called a Rayleigh quotient. The intermediate eigenvalues correspond to saddle points of .

These inequalities confine each eigenvalue of to the interval between two adjacent eigenvalues of ; the eigenvalues of are said to interlace those of . The following figure illustrates the case , showing a possible configuration of the eigenvalues of and of .

where the are nonnegative and sum to . If we greatly increase , the norm of the perturbation, then most of the increase in the eigenvalues is concentrated in the largest, since (5) bounds how much the smaller eigenvalues can change:

Exactly eigenvalues appear in one of these inequalities and appear in both. Therefore of the eigenvalues are equal to and so only eigenvalues can differ from . So perturbing the identity matrix by a Hermitian matrix of rank changes at most of the eigenvalues. (In fact, it changes exactly eigenvalues, as can be seen from a spectral decomposition.)

The theorem says that the eigenvalues of interlace those of for all . Two immediate implications are that (a) if is Hermitian positive definite then so are all its leading principal submatrices and (b) appending a row and a column to a Hermitian matrix does not decrease the largest eigenvalue or increase the smallest eigenvalue.

Since eigenvalues are unchanged under symmetric permutations of the matrix, the theorem can be reformulated to say that the eigenvalues of any principal submatrix of order interlace those of . A generalization to principal submatrices of order is given in the next result.

It follows by taking to be a unit vector in the formula that for all . And of course the trace of is the sum of the eigenvalues: . These relations are the first and last in a sequence of inequalities relating sums of eigenvalues to sums of diagonal elements obtained by Schur in 1923.

I suppose my question is really "why do we require norms in general to satisfy multiplicativity?". I ask this because for the usual absolute value on $\mathbb R$, I never feel like multiplicativity plays any "key role"; compare this to for example the omnipresent triangle inequality for the usual absolute value on $\mathbb R$, and in general in any metric space.

Other comments and answers have provided evidence that multiplicativity is an essential assumption in many results concerning $p$-adic norms. But if you think about how mathematical research is done, then the answer might be just the other way around.

Why do we require the $p$-adic norm to satisfy multiplicativity? Because researches in this direction turned out to produce many interesting and useful results, such as Ostrowski's theorem, while not much could be done without this requirement.

Well, if you loosen the restriction of multiplicativity to submultiplicativity, you get a far more intricate theory which seems to be not fully developed yet, but which (as @asahay points out in his answer and comments) connects naturally to nonarchimedean functional analysis (books by Bosch/Gntzer/Remmert, and Schneider), to Berkovich Spaces, and to Scholze's Perfectoid Spaces. See Kedlaya's paper On Commutative Nonarchimedean Banach Fields (arXiv link) for a good introduction.

As a first teaser, note that for any finite family (and many infinite families) $(\Vert \Vert_i :i \in I)$ of submultiplicative ring norms, their supremum is again a submultiplicative ring norm. I have an inkling that on $\mathbb Q$, all submultiplicative norms arise in this way from the multiplicative absolute values well-known via Ostrowski, but to be honest I'm not even in possession of a proof for that.

This is a seminar on formalising research level, MSc, or advanced undergraduate level mathematics in the Lean theorem prover. It is suitable for people who have some mathematical background, e.g. MSc students in a mathematics department. Speakers will assume some familiarity with the Lean but you certainly don't have to be a Lean expert

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