Given a positive matrix, in 1968 Gustafson proved
are eigenvalues of such that and are the largest and the smallest eigenvalues of respectively. Please see    .
The equality (1) played an important role in establishing what Gustafson calls “operator trigonometry”. In fact, for a positive matrix he defined to be
He proved (1) by using the convexity of the Hilbert space norm and other Hilbert space properties.
Later, in his investigation on problems of antieigenvalue theory, this author discovered a useful lemma which he calls the Two Nonzero Component Lemma or TNCL, for short (see    ). The antieigenvalue of an accretive operator acting on a complex Hilbert space is defined to be
For positive matrices, there is a relationship between the antieigenvalue of and. In a series of papers this author applied his TNCL to compute antieigenvalues of different types of operators, including normal operators. He also applied TNCL to compute other types of antieigenvalue quantities such as total antieigenvalues, higher order antieigenvalues,and joint antieigenvalues. Furthermore, he applied TNCL to solve some optimization problems in statistics, econometrics, and resource allocations. Please see  -  . Although this Lemma is implicitly used in all of the author's earlier papers up to 2008, it was not until 2008 that he stated a formal description of the Lemma in his paper titled, “Antieigenvalue Techniques in Statistics.” Below is the statement of the lemma. For an early proof of the lemma please see the author’s work in  .
Lemma 1 (The Two Nonzero Component Lemma) Let be the set of all sequences with nonnegative terms in the Banach Space. That is, let
be a function from to. Assume for, , and. Then the minimizing vectors for the function
on the convex set have at most two nonzero components.
What make the proof of the Lemma possible are the following two facts: First, the convexity of the set
Second, a special property that the functions
involved possess. If we set
then all restrictions of the form
have the same algebraic form as itself. For example if
then we have
which has the same algebraic form as
Indeed, for any,; all restrictions of the function
obtained by setting an arbitrary set of components of equal to zeros have the same algebraic form as. Obviously, not all functions have this property. For instance, for the function, , which does not have the same algebraic form as.
In the next section we prove that Gustafson’s identity (1) can be obtained using this author’s the Two Nonzero Component Lemma or TNCL. Our proof is elementary (comparing to Gustafson’s proof) in the sense that we use only TNCL and techniques of calculus.
2. A Proof of (1) Based on TNCL
Theorem 2 Let be a positive matrix where
are eigenvalues of such that and are the largest and the smallest eigenvalues of, then
Proof. Note that if we square the left hand side of (17) we get
Thus, we need to show
Now to follow notations usually used in differential calculus, let’s substitute with and consider
instead. With this change of notation. now we apply spectral theorem to the positive matrix and assume
are components of with respect to an orthogonal basis corresponding to
Therefore, we can rewrite (20) as
Applying TNCL we can assume any optimizing vector
is so that only two of its components, say and are nonzero and the rest of them are zero. Keeping that in mind, for such optimizing vectors (21) will be reduced to
To compute (22), let’s do some change of variables first. Substitute for, for, and for. (22) then becomes
For a fixed we compute the in (23) with respect to first. Consider the expression
We next find the derivative of (24) with respect to and set it equal to zero
and then solve it for. The solution is
Assume and note that the second derivative of (24) is
which is positive. This shows
is indeed a minimizing value. If we substitute from (26) in 24) and simplify we get
The derivative of (29) with respect to is
To find the optimizing value, we solve the following equation with respect to.
The solution of (31) is
If we substitute the value of from (32) in (29) and simplify we get
The second derivative of (29) is
which is negative, under our assumption that. This indicates that given by (32) is indeed a maximizing vector. Thus we have proved
Finally, we show that and. To show this note that
and notice that
Hence is decreasing and has the largest value when takes the smallest value. That is when and.
Remark 3 The equality (35) is valid even if is an infinite dimensional positive operator acting on a separable Hilbert space. The reason is that TNCL is valid both when has a finite or infinite number of components. However, in the case of an infinite dimensional positive operator, we do not know for what pair of and (35) holds.
We showed that TNCL can be used to prove an identity which was proved by Karl Gustafson in 1968. This identity was part of his min-max theorem. The identity was the basis of operator trigonometry. The original proof was based on Hilbert space techniques and convexity of operator norm. Using TNCM we reduced the problem to a very simple problem in elementary calculus. This indeed shows the power of this dimension reducing optimization lemma which is used by this author in many of his previous work. The lemma not only proved equality (1) but, as we noted in the remark above, it extended it to the case of positive operators on an infinite dimensional Hilbert space.
The author wishes to thank the referee of this paper for his helpful suggestions.