Cauchy-Schwarz Inequality: History, Applications & Example

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Knowing the largest something can be is a good thing and inequalities help us do this. In this lesson, we make sense of the often-used Cauchy-Schwarz inequality by providing some historical background, citing some applications, and doing an example.

Cauchy-Schwarz Inequality

Math relationships with equal signs (called equations) are very common. For example, the Pythagoras theorem is the equation a2 + b2 = c2. Perhaps not as common but still incredibly useful are math relationships with inequality signs like the Cauchy-Schwarz inequality.

Applications

Applications of this famous inequality include linear algebra (matrices, vectors, and transformations), probability theory (random variables, expected values, and correlation) as well as important topics in physics (uncertainty principle and photon noise) and engineering (root-mean-square values compared to peak values of a waveform). Using vectors, we will do an example showing how to verify the Cauchy-Schwarz inequality.

But first, the back-story.

History

Imagine growing up in France during the French Revolution. Such was the fate of Augustin-Louis Cauchy. In spite of these violent times, Cauchy learned mathematics and in 1821 published what would become the Cauchy-Schwarz inequality. But what about the other players in this famous result? Karl Hermann Amandus Schwarz, a former student of Cauchy, provided a proof of Cauchy's theory in 1888. And, the Russian mathematician Viktor Yakovlevich Bunyakovsky independently proved the theory many years earlier in 1859. Occasionally, this inequality includes all three names: the Cauchy-Bunyakovsky-Schwarz inequality.

So, what does this inequality look like?

Two Forms of the Cauchy-Schwarz Inequality

If we use vectors like x and y, the Cauchy-Schwarz Inequality looks like:


abs(x_dot_y)<=abs(x)abs(y)


In words, the length of the dot product between the vectors x and y is less than or equal to the product of the length of each vector.

Another form of the Cauchy-Schwarz Inequality says:


abs(x+y)<=abs(x)+abs(y)


This second form may be derived from the first by using right triangles and the Pythagoras theorem. The hypotenuse is x + y. This second form of Cauchy-Schwarz is saying the length of the hypotenuse is no bigger than the sum of the lengths of the other two sides. Otherwise, we could not have a triangle.

Vectors, lengths and dot products! How do we make sense of out all this? Easy, we do an example.

Example

Starting with x = (2, 3, -2) and y = (-1, 5, 6).


x=2i+3j-3k;y=-i+5k+6k


Can you relate the parentheses notation with the unit vectors? The two forms mean the same thing, x = (2, 3, -2) is just a short way to write the vector.

The ''length of x '' is | x |:


|x|=(2^2+3^2+(-1)^2)^.5=(4+9+4)^.5=(17)^.5=4.12


The answer is rounded to two decimal places. We can do the same with finding the length of y:


|y|=((-1)^2+5^2_6^2)^.5=(1+25+36)^.5=(62)^.5=7.87


What about the dot product?


x.y=2(-1)+3(5)+(-2)6=-2+15-12=1


In words, '' x dot y is the sum of products''. The products are

  • the i-component of x times the i-component of y
  • the j-component of x times the j-component of y
  • the k-component of x times the k-component of y

The dot product result is always a number. In our example, the length of this dot product is the absolute value of 1:


|x.y|=|1|=1


Just one more to do! Adding x and y:


x+y=(2+(-1))i+(3+5)j+((-2)+6)k=1i+8j+4k=i+8j+4k


Do you see how adding vectors is just adding the i-components, j-components, and k-components? Adding vectors produces another vector.

The length of x + y:


|x+y|=(1^2+8^2+4^2)^.5=(1+64+16)^.5=(81)^.5=9


Now, we go back to the Cauchy-Schwarz inequality. Is | xy | ≤ |x| |y| ? We calculate the left-hand side (LHS) and the right-hand side (RHS) separately:

LHS: | xy | = 1

RHS: |x| |y| = 4.12(7.87) = 32.42

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