From: Chris Hillman
Subject: Re: Geometric interpretation of the trace of a matrix?
Date: Fri, 5 Feb 1999 10:39:03 -0800
Newsgroups: sci.math,sci.physics,sci.physics.relativity
Keywords: Matrix inequalities
On 5 Feb 1999, Ilja Schmelzer wrote:
> Bruce Bowen writes:
> > Can anybody out there give a good geometric interpretation of the
> > trace of a matrix, i.e. the sum of the diagonal elements? For example,
> > in the case of the determinant, it is the (signed) volume of the
> > parallelepiped formed by all the rows (or columns) of a matrix.
>
> Once you like the volume as interpretation for determinant,
> the formula
>
> det|1+ xA| = 1 + x tr A + O(x^2)
>
> may be helpful. Thus, it is some type of derivative of the volume:
>
> tr A = d (det|1+ xA|)/dx
Uh oh, I can see confusion brewing! Ilja is saying that I + tA is the
"linearization" of P^t = exp tA, so the plain old derivative -at time
zero- of the volume change under P^t is
d/dt|(t=0) det P^t = d/dt|(t=0)
= d/dt|(t=0) det (I + tA)
= d/dt|(t=0) [1 + t (tr A) + O(t^2)]
= tr A
whereas I was saying that the -logarithmic derivative- of the volume
change at -any- time t is
d/dt log det P^t = d/dt log det exp t A
t trA
= d/dt log e
= d/dt t (trA)
= trA
So, the logarithmic derivative of the volume change under P^t is
-constant- (an invariant of the flow under P^t) and agrees with the
derivative at time t=0 of the volume change. Of course, (log V)' = V'/V
and V =1 at time t = 0 (remember we are talking about the volume change of
the progressively distorted unit cube), so this makes perfect sense!
Note that Ilja's polynomial evaluated at -t agrees with det(I - tA)
evaluated at t. It is interesting that the polynomial det(I - tA) arises
in invariant theory (Molien's theorem) and symbolic dynamics (dynamical
zeta functions). This in turn leads us to connections between the
principal eigenvalue (of suitable operators) and various notions of
entropy (Perron-Frobenius theory) and Lyapunov exponents in smooth
dynamical systems (Pesin formula, Osledec theorem) and statistical
mechanics (zeta functions again, and transfer operators).
See for instance my expository paper "Outline of the theory of G-sets" and
others available at
http://www.math.washington.edu/~hillman/papers.html
and also the textbook by Lind & Marcus, Introduction to Symbolic Dynamics
and Coding and the textbook by Katok and Hasselblatt, An Introduction to
the Modern Theory of Dynamical Systems, and expository papers by Baladi---
see her home page, which you can via my page
http://www.math.washington.edu/~hillman/symbolic.html
See also Cover and Thomas, Elements of Information Theory, for important
connections between:
1. the Brunn-Minkowski inequality
vol(S + T) >= vol(A + B)
where S, T are reasonably nice subsets of R^n and A,B are balls with vol S
= vol A, vol T = vol B.
2. the Shannon inequality
h(X+Y) >= h(U+V)
where X,Y are random variables and U,V are normal random variables with
h(U) = h(X) and h(V) = h(Y), where again normal variables are "symmetric".
3. Young's inequality from real analysis.
4. the Szasz inequality
P_1 >= P_2 >= P_3 >= ... P_n
where P_j is the -geometric mean- of the principal jxj minors of a
positive definite and symmetric but not neccessarily diagonal matrix K; in
particular P_1 >= P_n is the Hadamard inequality
product of diagonal entries >= det K
5. the inequality
S_1 >= S_2 >= ... S_n
where S_j is the -arithmetic mean- of the (1/j)-th powers of the principal
jxj minors of K; in particular S_1 >= S_n is the inequality
tr K (1/n)
---- >= (det K)
n
6. the Minkowksi inequality
[ det (K + L) ]^(1/n) >= (det K)^(1/n) + (det L)^(1/n)
where K,L are positive definite symmetric matrices.
7. lots more way cool stuff! :-)
Chris Hillman