Sum of squared logarithms - an inequality relating positive definite matrices and their matrix logarithm
© Bîrsan et al.; licensee Springer. 2013
Received: 21 January 2013
Accepted: 28 March 2013
Published: 12 April 2013
Let be such that and
This can also be stated in terms of real positive definite -matrices , : If their determinants are equal, , then
where log is the principal matrix logarithm and denotes the Frobenius matrix norm. Applications in matrix analysis and nonlinear elasticity are indicated.
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.
Thus, we obtain the following theorem.
and arrive at
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:
Thus, we have the equivalent formulation
Thus, Theorem 8 is also valid.
for any statements A, B, C, we can formulate the inequality (11) (i.e., Theorem 9) in the following equivalent manner.
We use the statement of Theorem 10 for the proof.
Therefore, our inequality (i.e., ) does not follow from majorization in disguise.
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.
which hold true since and . One can show in the same way that , , , so that (24) has been verified.
since the function is decreasing for .
since implies and .
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. □
The inequalities (32) and (33) are satisfied simultaneously if and only if , and .
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 , ). □
Then we have , and .
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 . □
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).
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 , .
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 , :
is automatically fulfilled.
so that the inequality (48) is equivalent to , i.e., we have to show that .
which means that since the function is increasing for . This completes the proof. □
because , and Theorem 6 provides the assertion. □
4 Some counterexamples for weakened assumptions
A counterexample for the two variable case can be constructed analogously.
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:
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 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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