This page will be about the Cauchy-Schwarz inequality. It is assumed that the reader is familiar with vector norms, and dot products from linear algebra.
Table Of Contents
- The History Of The Cauchy-Schwarz Inequality
- Review Of The Dot Product
- Arriving To The Cauchy-Schwarz Inequality
- The Cauchy-Schwarz Inequality
The History Of The Cauchy-Schwarz Inequality
The Cauchy-Schwarz inequality was named in honour of the French mathematician Augustin Cauchy and the German mathematician Hermann Schwarz. In this context, we are dealing with vectors but there are other variations of this inequality.
This inequality is sometimes called Cauchy’s inequality, the Schwarz inequality or the Bunyakovsky inequality. Bunyakovsky was a Russian mathematician who had his version of this equality 25 years before Schwarz.
Review Of The Dot Product
The dot product or the Euclidean inner product is an algebraic operation which takes two vectors of the same length and returns a scalar number.
Given the vectors and in ,
the dot product of and denoted by is:
If we have the special case of then we obtain the square of the norm as follows:
From the above result, we can identify that the norm or the length of a vector can be expressed as the square root of the dot product (below).
The dot product has an alternate formula for vectors and in . We have:
Arriving To The Cauchy-Schwarz Inequality
The dot product along with norms can help us find the cosine of an angle . A useful formula is:
where , are vectors in and is the angle between and . The angle is defined to be between 0 and ().
Recall from pre-calculus class that the cosine of an angle has a minimum of -1 and a maximum of +1. We can find the angle by taking the cosine inverse of .
From the above, we can note that we have the inequality:
Multiplying all the terms in the above inequality by gives:
We use the middle term and the one on the right as the focus for the Cauchy-Schwarz Inequality.
The Cauchy-Schwarz Inequality
Suppose there are two vectors and in , we have:
In terms of components, the above can be expressed as:
Recall that .
I do not know from the top of my head of the applications of this inequality. However, I would suspect that the Cauchy-Schwartz inequality does have useful mathematical applications in (higher-level) linear algebra, probability theory, real analysis, topology and vector algebra.