On the geometric meaning of the Cauchy Schwarz inequality, an intro to exterior powers, and surface integrals August 19, 2009
Posted by Phi. Isett in Uncategorized.Tags: Cauchy Schwarz, Cauchy Schwarz inequality, Determinants, Exterior Algebra, Exterior power, inner product, Surface integrals, Surface integration
3 comments
This is a brief remark on the Cauchy Schwarz inequality and one way of understanding its geometric meaning (at least in the context of a real inner product space). In a real inner product space , the inner product
allows for a generalization of intuitive geometric notions of “length”, “angle”, and “perpendicular” for vectors in
. For two elements
, I will write
to indicate the parallelogram formed by taking
and
as edges. The reader may as well simply imagine
for his favorite
and that the inner product is the usual dot product.
The Cauchy Schwarz inequality says that the area of a parallelogram is positive unless u and v are co-linear (it is also equivalent to the triangle inequality, but I will be talking about this formulation instead). If
and
are co-linear (they point in the same direction or perhaps opposite directions; maybe
) the parallelogram one forms with these two vectors is degenerate and has zero area; otherwise, you wind up with a parallelogram which has positive area. In fact, the volume of
is
when the two vectors
and
are perpendicular. If
and
fail to be perpendicular, then we can observe that shifting the edge
by any amount in the direction of the other edge
does not change the area of the parallelogram; thus the area of
is the same as the area
for any
. By choosing
to minimize the length of the first edge (pick
), we can make both edges perpendicular. The area of the resulting parallelogram must be non-negative (and must be positive when
and
are linearly independent), giving the inequality
. (This is actually a self-contained proof of the Cauchy-Schwartz inequality, and it’s the usual proof, but with some motivation regarding how to choose
.)
Of course, the Cauchy-Schwartz inequality is also equivalent to the triangle inequality, and the relationship between these two geometric interpretations can be seen by inspecting the parallelogram one forms by drawing ““.
I hope to write on how formalizing this area-of-parallelogram concept works (only on an intuitive level, for now). For those already familiar with exterior powers of vector spaces and exterior algebras (and their geometric meaning in terms of parallelograms), I can cut to the chase and say that there’s an induced inner product on the exterior powers and the Cauchy Schwartz inequality is what you get by writing .
I will also discuss the relation to surface integrals and determinants.