Exercise 3.2.1. a) Consider the vectors and where and are arbitrary positive real numbers. Use the Schwarz inequality involving and to derive a relationship between the arithmetic mean and the geometric mean .
b) Consider a vector from the origin to point , a second vector of length from to the point and the third vector from the origin to . Using the triangle inequality
derive the Schwarz inequality. (Hint: Square both sides of the inequality and expand the expression .)
Answer: a) From the Schwarz inequality we have
From the definitions of and , on the left side of the inequality we have
assuming we always choose the positive square root.
From the definitions of and we also have
so that the right side of the inequality is
again assuming we choose the positive square root. (We know is positive since both and are.)
The Schwartz inequality
or (dividing both sides by 2)
We thus see that for any positive real numbers and the geometric mean is less than the arithmetic mean .
b) From the triangle inequality we have
for the vectors and . Squaring the term on the left side of the inequality and using the commutative and distributive properties of the inner product we obtain
Squaring the term on the right side of the inequality we have
is thus equivalent to the inequality
Subtracting and from both sides of the inequality gives us
and dividing both sides of the inequality by 2 produces
Note that this is almost but not quite the Schwarz inequality: Since the Schwarz inequality involves the absolute value we must also prove that
(After all, the inner product might be negative, in which case the inequality would be trivially true, given that the term on the right side of the inequality is guaranteed to be positive.)
We have . Since the triangle inequality holds for any two vectors we can restate it in terms of and as follows:
Since squaring the term on the right side of the inequality produces
as it did previously. However squaring the term on the left side of the inequality produces
The original triangle inequality
is thus equivalent to
Since we have both and we therefore have
which is the Schwarz inequality.
So the triangle inequality implies the Schwarz inequality.
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.