Cauchy's Inequality

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Theorem

$\displaystyle \sum {r_i^2} \sum {s_i^2} \ge \left({\sum {r_i s_i}}\right)^2$

where all of $r_i, s_i \in \R$.


Proof 1

For any $\lambda \in \R$, we define $f: \R \to \R$ as the function:

$\displaystyle \map f \lambda = \sum {\paren {r_i + \lambda s_i}^2}$

Now:

$\map f \lambda \ge 0$

because it is the sum of squares of real numbers.


Hence:

\(\displaystyle \forall \lambda \in \R: \map f \lambda\) \(\equiv\) \(\displaystyle \sum {\paren {r_i^2 + 2 \lambda r_i s_i + \lambda^2 s_i^2} } \ge 0\)
\(\displaystyle \) \(\equiv\) \(\displaystyle \sum {r_i^2} + 2 \lambda \sum {r_i s_i} + \lambda^2 \sum {s_i^2} \ge 0\)

This is a quadratic equation in $\lambda$.

From Solution to Quadratic Equation:

$\displaystyle a \lambda^2 + b \lambda + c = 0: a = \sum {s_i^2}, b = 2 \sum {r_i s_i}, c = \sum {r_i^2}$


The discriminant of this equation (that is $b^2 - 4 a c$) is:

$D := \displaystyle 4 \paren {\sum {r_i s_i} }^2 - 4 \sum {r_i^2} \sum {s_i^2}$


Aiming for a contradiction, suppose $D$ is (strictly) positive.

Then $\map f \lambda = 0$ has two distinct real roots, $\lambda_1 < \lambda_2$, say.

From Sign of Quadratic Function Between Roots, it follows that $f$ is (strictly) negative somewhere between $\lambda_1$ and $\lambda_2$.

But we have:

$\forall \lambda \in \R: \map f \lambda \ge 0$


From this contradiction it follows that:

$D \le 0$

which is the same thing as saying:

$\displaystyle \sum {r_i^2} \sum {s_i^2} \ge \paren {\sum {r_i s_i} }^2$

$\blacksquare$


Proof 2

From the Complex Number form of the Cauchy-Schwarz Inequality, we have:

$\displaystyle \sum \left|{w_i}\right|^2 \left|{z_i}\right|^2 \ge \left|{\sum w_i z_i}\right|^2$

where all of $w_i, z_i \in \C$.


As elements of $\R$ are also elements of $\C$, it follows that:

$\displaystyle \sum \left|{r_i}\right|^2 \left|{s_i}\right|^2 \ge \left|{\sum r_i s_i}\right|^2$

where all of $r_i, s_i \in \R$.


But from the definition of modulus, it follows that:

$\displaystyle \forall r_i \in \R: \left|{r_i}\right|^2 = r_i^2$


Thus:

$\displaystyle \sum {r_i^2} \sum {s_i^2} \ge \left({\sum {r_i s_i}}\right)^2$

where all of $r_i, s_i \in \R$.

$\blacksquare$


Also known as

This result is also known as the Cauchy-Schwarz Inequality, although that name is also given to the more general Cauchy-Bunyakovsky-Schwarz Inequality.


Source of Name

This entry was named for Augustin Louis Cauchy.


Historical Note

The result known as Cauchy's Inequality was first published by Augustin Louis Cauchy in $1821$.

It is a special case of the Cauchy-Bunyakovsky-Schwarz Inequality.


Sources