Abstract
Keywords:
matrix logarithm; elementary symmetric polynomials; inequality; characteristic polynomial; positive definite matrices; means1 Introduction
Convexity is a powerful source for obtaining new inequalities; see, e.g., [1,2]. In applications coming from nonlinear elasticity, we are faced, however, with variants of the squared logarithm function; see the last section. The function is neither convex nor concave. Nevertheless, the sum of squared logarithms inequality holds. We will proceed as follows: In the first section, we will give several equivalent formulations of the inequality, for example, in terms of the coefficients of the characteristic polynomial (Theorem 1), in terms of elementary symmetric polynomials (Theorem 3), in terms of means (Theorem 5) or in terms of the Frobenius matrix norm (Theorem 7). A proof of the inequality will be given in Section 2, and some counterexamples for slightly changed variants of the inequality are discussed in Section 3. In the last section, an application of the sum of squared logarithms inequality in matrix analysis and in the mathematical theory of nonlinear elasticity is indicated.
2 Formulations of the problem
All theorems in this section are equivalent.
Theorem 1Fororletbe positive definite real matrices. Let the coefficients of the characteristic polynomials ofandsatisfy
Then
For , we will now give equivalent formulations of this statement. The case can be treated analogously. For its proof, see Remark 15. By orthogonal diagonalization of and , the inequalities can be rewritten in terms of the eigenvalues , , and , , , respectively.
Theorem 2Let the real numbersandbe such that
Then
The elementary symmetric polynomials, see, e.g., [[3], p.178]
are known to have the Schurconcavity property (i.e., is Schurconvex) [1,4]; see (16). It is possible to express the problem in terms of these elementary symmetric polynomials as follows.
Then
Thus, we obtain the following theorem.
Theorem 4Let the real numbersandbe such that
Then
The conditions (3) are also simple expressions in terms of arithmetic, harmonic and geometric and quadratic mean
Theorem 5Letand. Then, (‘reverse!’) andimply
We denote by
and arrive at
Theorem 6Let the real numbersandbe such that, and
Then
If we again view and as eigenvalues of positive definite matrices, an equivalent formulation of the problem can be given in terms of their Frobenius matrix norms:
Theorem 7For, letbe positive definite real matrices. Let
Then
Let us reconsider the formulation from Theorem 5. If we denote
Theorem 8Let the real numbers, , and, , be such that
Then
In order to prove Theorem 8, one can assume without loss of generality that
Thus, we have the equivalent formulation
Theorem 9Let the real numbers, , and, , be such that
Then
Let us prove that Theorem 8 can be reformulated as Theorem 9. Indeed, let us assume that Theorem 9 is valid and show that the statement of Theorem 8 also holds true. We denote by s the sum and we designate
Then the real numbers and satisfy the hypotheses of Theorem 9 and we obtain . This inequality is equivalent to
which, by virtue of the condition (7)_{3}, reduces to
Thus, Theorem 8 is also valid.
By virtue of the logical equivalence
for any statements A, B, C, we can formulate the inequality (11) (i.e., Theorem 9) in the following equivalent manner.
Theorem 10Let the real numbers, , and, , be such that
Then one of the following inequalities holds:
We use the statement of Theorem 10 for the proof.
Before continuing, let us show that our new inequality is not a consequence of majorization and Karamata’s inequality [5]. Consider and arranged already in decreasing order and . If
we say that z majorizes c, denoted by . The following result is well known [[6], p.89], [4,5]. If is convex, then
is called Schurconvex. In Theorem 8, the convex function to be considered would be . Do conditions (7) (upon rearrangement of if necessary) yield already majorization ? This is not the case, as we explain now. Let the real numbers and be such that
These conditions do not imply the majorization ,
Therefore, our inequality (i.e., ) does not follow from majorization in disguise.
Indeed, let
and
Then we have and , together with
but the majorization inequalities (18) are not satisfied, since .
3 Proof of the inequality
Of course, we may assume without loss of generality that and (and the same for , , , ).
The proof begins with the crucial lemma.
Lemma 11Let the real numbersandbe such that
Then the inequality
is satisfied if and only if the relation
holds, or equivalently, if and only if
holds.
Proof Let us denote by . Then, from (19), it follows
and we find
In view of (19) and , , one can show that
Indeed, let us verify the relations (24). We have
which hold true since and . Similarly, we have
which holds true since and . Also, we have
which hold true since and . One can show in the same way that , , , so that (24) has been verified.
We prove now that the inequality (21) holds if and only if (22) holds. Indeed, using (23)_{2,4} and (24) we get
since the function is decreasing for .
Let us prove next that the inequalities (20) and (21) are equivalent. To accomplish this, we introduce the function by
Taking into account (23) and (24)_{1}, the inequality (20) can be written equivalently as
which is equivalent to
since the function f defined by (25) is monotone increasing on , as we show next. To this aim, we denote by
Then the function (25) can be written as
We have to show that is decreasing with respect to . We compute the first derivative
The function (28) has the same sign as the function
In order to show that for all , we remark that for fixed and we compute
Consequently, the function is decreasing with respect to r and for any we have that
From (29) and (31), it follows that is decreasing with respect to . This means that is increasing as a function of , i.e., the relation (26) is indeed equivalent to and the proof is complete. □
Consequence 12Let the real numbersandbe such that
Then one of the following inequalities holds:
or
The inequalities (32) and (33) are satisfied simultaneously if and only if, and.
Proof According to Lemma 11, the inequality (32) is equivalent to
while the inequality (33) is equivalent to
Since one of the relations (34) and (35) must hold, we have proved that one of the inequalities (32) and (33) is satisfied. They are simultaneously satisfied if and only if both (34) and (35) hold true, i.e., (and consequently , ). □
Consequence 13Let the real numbersandbe such that
Proof Since by hypothesis holds, we can apply Lemma 11 to deduce and .
On the other hand, by virtue of the inverse inequality and Lemma 11, we obtain and . In conclusion, we get , and . □
Proof of Theorem 10 In order to prove (13), we define the real numbers
Then we have
If we apply the Consequence 12 for the numbers and , then we obtain that
In what follows, let us show that
we have and . With the help of the function h defined in (27), we can write the inequality (39) in the form
The relation (40) asserts that the function h defined in (27) is increasing with respect to the first variable . To show this, we compute the derivative
By virtue of the Chebyshev’s sum inequality, we deduce from (41) that
Indeed, the Chebyshev’s sum inequality [[6], 2.17] asserts that: if and then
In our case, we derive the following result: for any real numbers x, y, z such that , the inequality
holds true, with equality if and only if .
Applying the result (43) to the function (41), we deduce the relation (42). This means that is an increasing function of r, i.e. the inequality (40) holds, and hence, we have proved (39).
One can show analogously that the inequality
is also valid. From (38), (39) and (44), it follows that the assertion (13) holds true. Thus, the proof of Theorem 10 is complete. □
Since the statements of the Theorems 8 and 10 are equivalent, we have proved also the inequality (8).
Remark 14 The inequality (8) becomes an equality if and only if , .
Proof Indeed, assume that . Then we can apply the Consequence 12 and we deduce that
Taking into account (7)_{1,2} in conjunction with (45), we find
By virtue of (46), we can apply the Consequence 13 to derive , and consequently , . □
Let us prove the following version of the inequality (6) for two pairs of numbers , and , :
Remark 15 If the real numbers and are such that
then the inequality
holds true. Note that the additional condition
is automatically fulfilled.
Proof Since and , , we have , and
so that the inequality (48) is equivalent to , i.e., we have to show that .
Indeed, if we insert and into the inequality (47)_{1} then we find
which means that since the function is increasing for . This completes the proof. □
Alternative proof of Remark 15 Let . Then (47) implies and as well as
4 Some counterexamples for weakened assumptions
Example 16 Unlike in the 2D case in Remark 15, for two triples of numbers the second condition (18)_{2} of Theorem 2, namely , cannot be removed. Let
but
Example 17 The condition cannot be weakened to . Indeed, let , , . Then
But nevertheless
A counterexample for the two variable case can be constructed analogously.
Example 18 Even with an analogous condition, the inequality (4) does not hold for numbers (without further assumptions). Indeed, let
Furthermore,
and
Since , we have and, therefore,
Nevertheless, for the sum of squared logarithms, the ‘reverse’ inequality
holds true.
Example 19 The inequality (4) does not remain true either, if the function is replaced by its linearization . Indeed, let , , , , , . Then
and
But
5 Conjecture for arbitrary n
The structure of the inequality in dimensions and and extensive numerical sampling strongly suggest that the inequality holds for all if the n corresponding conditions are satisfied. More precisely, in terms of the elementary symmetric polynomials, we expect the following:
Conjecture 20Letandfor. If for allwe have
then
6 Applications
The investigation in this paper has been motivated by some recent applications. The new sum of squared logarithms inequality is one of the fundamental tools in deducing a novel optimality result in matrix analysis and the conditions in the form (3) had been deduced in the course of that work. Optimality in the matrix problem suggested the sum of squared logarithms inequality. Indeed, based on the present result in [7], it has been shown that for all invertible and for any definition of the matrix logarithm as possibly multivalued solution of it holds
where is the Hermitian part of and is the unitary factor in the polar decomposition of Z into unitary and Hermitian positive definite matrix H
This result (50) generalizes the fact that for any complex logarithm and for all
The optimality result (50) can now also be viewed as another characterization of the unitary factor in the polar decomposition. In addition, in a forthcoming contribution [8], we use (50) to calculate the geodesic distance of the isochoric part of the deformation gradient to in the canonical leftinvariant Riemannian metric on , to the effect that
where is the orthogonal projection of to trace free matrices. Thereby, we provide a rigorous geometric justification for the preferred use of the Henckystrain measure in nonlinear elasticity and plasticity theory [9].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed fully to all parts of the manuscript. Notably all ideas have emerged by continuous discussions among them.
Acknowledgements
The first author (MB) was supported by the German state grant: ‘Programm des Bundes und der Länder für bessere Studienbedingungen und mehr Qualität in der Lehre’.
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