_{1}

^{*}

In this paper we show that the author’s Two Nonzero Lemma (TNCL) can be applied to present a simple proof for a very useful equality which was first proved by Karl Gustafson in 1968. Gustafson used Hilbert space methods, including convexity of the Hilbert space norm, to prove this identity which was the basis of his matrix trigonometry. By applying TNCL, we will reduce the problem to a simple problem of ordinary calculus.

Given a positive matrix

where

are eigenvalues of

The equality (1) played an important role in establishing what Gustafson calls “operator trigonometry”. In fact, for a positive matrix

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 [

For positive matrices, there is a relationship between the antieigenvalue of

Lemma 1 (The Two Nonzero Component Lemma) Let

Let

be a function from

on the convex set

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

of

have the same algebraic form as

then we have

which has the same algebraic form as

Indeed, for any

obtained by setting an arbitrary set of

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.

Theorem 2 Let

are eigenvalues of

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

instead. With this change of notation. now we apply spectral theorem to the positive matrix

are components of

Therefore, we can rewrite (20) as

Applying TNCL we can assume any optimizing vector

is so that only two of its components, say

To compute (22), let’s do some change of variables first. Substitute

For a fixed

We next find the derivative of (24) with respect to

and then solve it for

Assume

which is positive. This shows

is indeed a minimizing value. If we substitute

The derivative of (29) with respect to

To find the optimizing value

The solution of (31) is

If we substitute the value of

The second derivative of (29) is

which is negative, under our assumption that

Finally, we show that

Now define

and notice that

Hence

Remark 3 The equality (35) is valid even if

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.

Seddighin, M. (2017) Two Nonzero Component Lemma and Matrix Trigonometry. Advances in Linear Algebra & Matrix Theory, 7, 1-6. https://doi.org/10.4236/alamt.2017.71001