MSC: 26D05, 26D07.
Keywords:matrix logarithm; elementary symmetric polynomials; inequality; characteristic polynomial; positive definite matrices; means
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.
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.
The elementary symmetric polynomials, see, e.g., [, p.178]
Thus, we obtain the following theorem.
The conditions (3) are also simple expressions in terms of arithmetic, harmonic and geometric and quadratic mean
We denote by
and arrive at
Let us reconsider the formulation from Theorem 5. If we denote
In order to prove Theorem 8, one can assume without loss of generality that
Thus, we have the equivalent formulation
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
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.
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 . Consider and arranged already in decreasing order and . If
is called Schur-convex. 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
3 Proof of the inequality
The proof begins with the crucial lemma.
Then the inequality
is satisfied if and only if the relation
holds, or equivalently, if and only if
and we find
Indeed, let us verify the relations (24). We have
We prove now that the inequality (21) holds if and only if (22) holds. Indeed, using (23)2,4 and (24) we get
Taking into account (23) and (24)1, the inequality (20) can be written equivalently as
which is equivalent to
Then the function (25) can be written as
The function (28) has the same sign as the function
Then one of the following inequalities holds:
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 , ). □
Proof of Theorem 10 In order to prove (13), we define the real numbers
Then we have
In what follows, let us show that
By virtue of the Chebyshev’s sum inequality, we deduce from (41) that
Indeed, the Chebyshev’s sum inequality [, 2.17] asserts that: if and then
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).
Taking into account (7)1,2 in conjunction with (45), we find
then the inequality
holds true. Note that the additional condition
is automatically fulfilled.
4 Some counterexamples for weakened assumptions
A counterexample for the two variable case can be constructed analogously.
Nevertheless, for the sum of squared logarithms, the ‘reverse’ inequality
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:
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 , it has been shown that for all invertible and for any definition of the matrix logarithm as possibly multivalued solution of it holds
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 , we use (50) to calculate the geodesic distance of the isochoric part of the deformation gradient to in the canonical left-invariant 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 Hencky-strain measure in nonlinear elasticity and plasticity theory .
The authors declare that they have no competing interests.
All authors contributed fully to all parts of the manuscript. Notably all ideas have emerged by continuous discussions among them.
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’.
Neff, P, Nagatsukasa, Y, Fischle, A: The unitary polar factor Q=Up minimizes ∥Log(Q∗Z)∥2 and ∥sym∗Log(Q∗Z)∥2 in the spectral norm in any dimension and the Frobenius matrix norm in three dimensions (2013, submitted)
Neff, P, Eidel, B, Osterbrink, F, Martin, R: The isotropic Hencky strain energy ∥logU∥2 measures the geodesic distance of the deformation gradient F∈GL+(n) to SO(n) in the unique left-invariant Riemannian metric on GL+(n) which is also right O(n)-invariant (2013, in preparation)