Page 293 - Matrix Analysis & Applied Linear Algebra

5.3 Inner-Product Spaces 289
Example 5.3.3

To illustrate the utility of the ideas presented above, consider the proposition

T 2 T T
trace A B ≤ trace A A trace B B for all A, B ∈ m×n .

Problem: How would you know to formulate such a proposition and, second,
how do you prove it?

Solution: The answer to both questions is the same. This is the CBS inequality

m×n T
in equipped with the standard inner product A B = trace A B and

T
associated norm A = A A = trace (A A)because CBS says
F
2 2 2 T 2 T T
A B ≤ A B F =⇒ trace A B ≤ trace A A trace B B .
F
The point here is that if your knowledge is limited to elementary matrix manip-
ulations (which is all that is needed to understand the statement of the propo-
sition), formulating the correct inequality might be quite a challenge to your
intuition. And then proving the proposition using only elementary matrix ma-
nipulations would be a significant task—essentially, you would have to derive a
version of CBS. But knowing the basic facts of inner-product spaces makes the
proposition nearly trivial to conjecture and prove.

Since each inner product generates a norm by the rule = , it’s
natural to ask if the reverse is also true. That is, for each vector norm
on a space V, does there exist a corresponding inner product on V such that
2
= ?If not, under what conditions will a given norm be generated by
an inner product? These are tricky questions, and it took the combined efforts
38
of Maurice R. Fr´echet (1878–1973) and John von Neumann (1903–1957) to
provide the answer.

38
Maurice Ren´eFr´echet began his illustrious career by writing an outstanding Ph.D. dissertation
in 1906 under the direction of the famous French mathematician Jacques Hadamard (p. 469)
in which the concepts of a metric space and compactness were first formulated. Fr´echet devel-
oped into a versatile mathematical scientist, and he served as professor of mechanics at the
University of Poitiers (1910–1919), professor of higher calculus at the University of Strasbourg
(1920–1927), and professor of differential and integral calculus and professor of the calculus of
probabilities at the University of Paris (1928–1948).
Born in Budapest, Hungary, John von Neumann was a child prodigy who could divide eight-
digit numbers in his head when he was only six years old. Due to the political unrest in
Europe, he came to America, where, in 1933, he became one of the six original professors
of mathematics at the Institute for Advanced Study at Princeton University, a position he
retained for the rest of his life. During his career, von Neumann’s genius touched mathematics
(pure and applied), chemistry, physics, economics, and computer science, and he is generally
considered to be among the best scientists and mathematicians of the twentieth century.