Two Nonzero Component Lemma and Matrix Trigonometry

Show more

1. Introduction

Given a positive matrix, in 1968 Gustafson proved

(1)

where

(2)

are eigenvalues of such that and are the largest and the smallest eigenvalues of respectively. Please see [1] [2] [3] .

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

(3)

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 [4] [5] [6] ). The antieigenvalue of an accretive operator acting on a complex Hilbert space is defined to be

(4)

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 [5] - [11] . 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 [5] .

Lemma 1 (The Two Nonzero Component Lemma) Let be the set of all sequences with nonnegative terms in the Banach Space. That is, let

(5)

Let

(6)

be a function from to. Assume for, , and. Then the minimizing vectors for the function

(7)

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

(8)

Second, a special property that the functions

(9)

involved possess. If we set

(10)

then all restrictions of the form

(11)

of

(12)

have the same algebraic form as itself. For example if

(13)

then we have

(14)

which has the same algebraic form as

(15)

Indeed, for any,; all restrictions of the function

(16)

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

(17)

Proof. Note that if we square the left hand side of (17) we get

(18)

Thus, we need to show

(19)

Now to follow notations usually used in differential calculus, let’s substitute with and consider

(20)

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

(21)

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

(22)

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

(23)

For a fixed we compute the in (23) with respect to first. Consider the expression

(24)

We next find the derivative of (24) with respect to and set it equal to zero

(25)

and then solve it for. The solution is

(26)

Assume and note that the second derivative of (24) is

(27)

which is positive. This shows

(28)

is indeed a minimizing value. If we substitute from (26) in 24) and simplify we get

(29)

The derivative of (29) with respect to is

(30)

To find the optimizing value, we solve the following equation with respect to.

(31)

The solution of (31) is

(32)

If we substitute the value of from (32) in (29) and simplify we get

(33)

The second derivative of (29) is

(34)

which is negative, under our assumption that. This indicates that given by (32) is indeed a maximizing vector. Thus we have proved

(35)

Finally, we show that and. To show this note that

(36)

Now define

(37)

and notice that

(38)

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.

3. Conclusion

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.

Acknowledgement

The author wishes to thank the referee of this paper for his helpful suggestions.

References

[1] Gustafson, K. (2010) On My Min-Max Theorem (1968) and Its Consequences. Acta et Commentationes Universitatis Tartuensis de Mathematica, 14, 45-51.

[2] Gustafson, K. (1968) A Min-Max Theorem. Notices of the American Mathematical Society, 15, 799.

[3] Gustafson, K. and Rao, D. (1997) Numerical Range. Springer.

https://doi.org/10.1007/978-1-4613-8498-4

[4] Gustafson, K. and Seddighin, M. (1989) Antieigenvalue Bounds. Journal of Mathematical Analysis and Applications, 143, 327-340.

https://doi.org/10.1016/0022-247X(89)90044-9

[5] Seddighin, M. (2002) Antieigenvalues and Total Antieigenvalues of Normal Operators. Journal of Mathematical Analysis and Applications, 274, 239-254.

https://doi.org/10.1016/S0022-247X(02)00295-0

[6] Seddighin, M. (2009) Antieigenvalue Techniques in Statistics. Linear Algebra and Its Applications, 430, 2566-2580.

https://doi.org/10.1016/j.laa.2008.05.007

[7] Seddighin, M. and Gustafsib, K. (2005) On the Eigenvalues which Express Antieigenvalues. International Journal of Mathematics and Mathematical Sciences, 2005, 1543-1554.

[8] Seddighin, M. (2010) Gustafson K. Slant Antieigenvalues and Slant Antieigenvectors of Operators. Journal of Linear Algebra and Applications, 432, 1348-1362.

https://doi.org/10.1016/j.laa.2009.11.001

[9] Seddighin, M. (2011) Slant Joint Antieigenvalues and Antieigenvectors of Operators in Normal Subalgebras. Journal of Linear Algebra and its Applications, 434, 1395-1408.

https://doi.org/10.1016/j.laa.2010.11.020

[10] Seddighin, M. (2014) Proving and Extending Greub-Reinboldt Inequality Using the Two Nonzero Component Lemma. Advances in Linear Algebra & Matrix Theory, 4, 120-127.

https://doi.org/10.4236/alamt.2014.42010

[11] Seddighin, M. (2014) Application of the Two Nonzero Component Lemma in Resource Allocation. Journal of Applied Mathematics and Physics, 2, 653-661.