Cauchy-Schwarz Inequality
In algebra, the Cauchy-Schwarz Inequality, also known as the Cauchy–Bunyakovsky–Schwarz Inequality or informally as Cauchy-Schwarz, is an inequality with many ubiquitous formulations in abstract algebra, calculus, and contest mathematics. In high-school competitions, its applications are limited to elementary and linear algebra.
Its elementary algebraic formulation is often referred to as Cauchy's Inequality and states that for any list of reals and
,
with equality if and only if there exists a constant
such that
for all
, or if every number in one of the lists is zero. Along with the AM-GM Inequality, Cauchy-Schwarz forms the foundation for inequality problems in intermediate and olympiad competitions. It is particularly crucial in proof-based contests.
Its linear algebraic forulation states that for any vectors and
in
,
with equality if and only if there exists a scalar
such that
, or if one of the vectors is zero. This formulation comes in handy in linear algebra problems at intermediate and olympiad problems.
The full Cauchy-Schwarz Inequality is written in terms of abstract vector spaces. Under this formulation, the elementary algebraic, linear algebraic, and calculus formulations are different cases of the general inequality.
Contents
Proof
Consider the vectors and
. If
is the angle formed by
and
, then the left-hand side of the inequality is equal to the square of the dot product of
and
, or
.The right hand side of the inequality is equal to
. The inequality then follows from
, with equality when one of
is a multiple of the other, as desired.
Lemmas
Complex Form
The inequality sometimes appears in the following form.
Let and
be complex numbers. Then
This appears to be more powerful, but it follows from
General Form
Let be a vector space, and let
be an inner product. Then for any
,
with equality if and only if there exist constants
not both zero such that
.
Proof 1
Consider the polynomial of
This must always be greater than or equal to zero, so it must have a non-positive discriminant, i.e.,
must be less than or equal to
, with equality when
or when there exists some scalar
such that
, as desired.
Proof 2
We consider
Since this is always greater than or equal to zero, we have
Now, if either
or
is equal to
, then
. Otherwise, we may normalize so that
, and we have
with equality when
and
may be scaled to each other, as desired.
Proof 3
Consider for some scalar
. Then:
(by the Trivial Inequality)
.
Now, let
. Then, we have:
.
Problems
Introductory
- Consider the function
, where
is a positive integer. Show that
. (Source)
- (APMO 1991 #3) Let
,
,
,
,
,
,
,
be positive real numbers such that
. Show that
Intermediate
- Let
be a triangle such that
where
and
denote its semiperimeter and inradius, respectively. Prove that triangle
is similar to a triangle
whose side lengths are all positive integers with no common divisor and determine those integers.
(Source)
Olympiad
is a point inside a given triangle
.
are the feet of the perpendiculars from
to the lines
, respectively. Find all
for which
is least.
(Source)
Other Resources
Books
- The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities by J. Michael Steele.
- Problem Solving Strategies by Arthur Engel contains significant material on inequalities.