Difference between revisions of "Cauchy-Schwarz Inequality"
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== Contest Problem Solving == | == Contest Problem Solving == | ||
This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]]. | This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the [[United States of America Mathematics Olympiad | USAMO]] and [[International Mathematics Olympiad | IMO]]. | ||
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+ | == Other Resources == | ||
+ | * [http://en.wikipedia.org/wiki/Cauchy-Schwartz_inequality Wikipedia entry] |
Revision as of 02:40, 18 June 2006
The Cauchy-Schwarz Inequality (which is known by other names, including Cauchy's Inequality) states that, for two sets of real numbers and
, the following inequality is always true:
Equality holds if and only if .
Proof
There are many ways to prove this; one of the more well-known is to consider the equation
.
Expanding, we find the equation to be of the form
where ,
, and
. By the Trivial Inequality, we know that the left-hand-side of the original equation is always at least 0, so either both roots are complex numbers, or there is a double root at
. Either way, the discriminant of the equation is nonpositive. Taking the discriminant,
and substituting the above values of A, B, and C leaves us with the Cauchy-Schwarz Inequality,
or, in the more compact sigma notation,
Note that this also gives us the equality case; equality holds if and only if the discriminant is equal to 0, which is true if and only if the equation has 0 as a double root, which is true if and only if .
Contest Problem Solving
This inequality is used very frequently to solve Olympiad-level Inequality problems, such as those on the USAMO and IMO.