ELEMENTARY GEOMETRY IN R^n


CONVEX POLYTOPES  (closely related to linear algebra)

The "convex polytope" with vertices v1, v2,..., vm, is the set of all
points of the form

      SUMMATION ci vi,   where each ci >= 0 and  SUMMATION ci = 1

Here the v's (written here as vi rather than the more precise, but cumbersome,
v[i]) are n-dimensional real vectors and the c's are real scalars.

The unit cube (also called "hypercube") with one vertex at the origin
is the convex polytope with 2^n vertices, all of whose coordinates assume
only the values 0 and/or 1.  Its volume is 1.  Its "diameter" is the distance
between a pair of maximally distant vertices.  This is easily seen to be
n^(1/2).  The center of this cube is 1/2 (1,1,1,...,1)^T.  If this center
is translated to the origin, then the vertices would all have
coordinates whose values are 1/2 and/or -1/2.  The distance from the center
to any vertex is called the "outside radius".  Its value is n^(1/2)/2.
The distance from the center to any face is called the "inside radius".
Its value is 1/2.

A hyper-pyramid can be obtained by slicing off one corner of this cube.
In the original (untranslated) version, the vertex of the corner is at 0.
The pyramid is a simplex which also includes n other vertices, each of
which is located at a unit vector.  The volume of this simplex is 1/n!
The "face" of this hyper-pyramid opposite the origin is a REGULAR simplex
whose n vertices are the n unit vectors in n dimensions. All edges of this
simplex have length 2^(1/2).  The center of this simplex lies at 1/n
(1,1,1,...1)^T.  The outside radius of this regular simplex is n-1
times the inside radius.  If the simplex is translated to be recentered
at the origin, then the sum of its n vertices is 0, from which it is seen
that the simplex now lies in an n-1 dimensional subspace of R^n.

   CONCLUSION:  The ratio of the outside radius to the inside radius of a
   typical polytope is large when the dimensionality is large.

   PROBLEMS:  What are the "altitudes" and "bases" of the pyramid and the
    regular simplex?   Verify that each altitude is perpendicular to the
    corresponding base.


SPHERES (more related to calculus then to linear algebra)

Using appropriate techniques from integral calculus, it can be shown that
the volume of the sphere of radius R in n dimensions is

      pi^(n/2) R^n / (n/2)!

where pi = 3.14159... and the factorial function of k is defined as

      k! = INTEGRAL t^k exp(-t) dt             (1)

where the integral on t runs from 0 to infinity.

Integration of (1) by parts yields the basic factorial recursion for
positive k, namely

      k! = k (k-1)!

This recursion does NOT require k to be an integer.
For integers, repeated use of this recursion can reduce the value to 0!
For half-integers, repeated use of this recursion can reduce the value
to (-1/2)!  Evaluation of the definite integrals of (1) reveals that

     0! = 1
and
     (-1/2)! = pi^(1/2)

Equation (1) can also be used to obtain a good asymptotic estimate of k!
when k is very large, namely

     k! ~  k^k  e^(-k)  (2 pi k)^(1/2) (1 + 1/(12 k) + ... )

This asymptotic expansion is known as Stirling's formula.

   CONCLUSION: For large n, most of the volume of an n-dimensional sphere
    is located very near its surface.  Most of the volume of an n-dimensional
    polytope is located near its vertices (and far from the centers of its
    faces).

   PROBLEM:  Let n be very large.  A sphere and a cube, both centered at the
    origin, have the same volume.  What is the ratio of their inside radii?
    What is the ratio of their outside radii?


SIGNS OF PERMUTATIONS

Since many students have some troubles with signs of permutations, wherever
they are first encountered, I tried to exemplify them in detail on
October 7.  The main points are actually quite simple, namely:

  Of the n! permutations, some are even and others are odd.
  Odd permutations are associated with -1; even permutations, with +1.  
  Parities of permutations compose as parities of numbers add, which is
   the same as -1 and +1 multiply. (because -1 is an odd power of -1 and
   +1 is an even power of -1).
  The identity permutation is even.
  The parity of a permutation is the same as the parity of its inverse.
  Any adjacent transposition is odd.
  Any transposition is odd.
  Any permutation consisting of a single cycle of even length is odd.
  Any permutation consisting of a single cycle of odd length is even.

Using parities of permutations and multi-linearity, we can express
the value of a determinant as the signed sum of n! terms, each of which
is the signed product of a set of n components of the matrix which correspond
to a permutation matrix.  This shows that the value of the determinant lies
in the same ring as its entries; in particular, the determinant of a matrix
with integer entries must be an integer.  Even though an LU factorization
of such a matrix will typically yield fractional entries, if the diagonal
of L is all ones, and the diagonal of U is some set of fractions, then in the
product of all of the diagonal entries of U (which equals the value of the
determinant) the fractions must cancel out to become an integer.

This result also facilitates the proofs of other principal properties of
determinants, as in the book.


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