Sum of squared logarithms - An inequality relating positive definite matrices and their matrix logarithm

TL;DR

This paper establishes a new inequality relating the sums of squared logarithms of positive variables and extends it to matrix analysis, linking determinants, traces, and matrix logarithms with applications in elasticity.

Contribution

It introduces a novel inequality connecting positive definite matrices' determinants, traces, and their matrix logarithms, with implications for matrix analysis and nonlinear elasticity.

Findings

Proves an inequality for positive variables with equal products and sums.

Extends the inequality to positive definite matrices involving determinants and traces.

Highlights applications in matrix analysis and nonlinear elasticity.

Abstract

Let y1, y2, y3, a1, a2, a3 > 0 be such that y1 y2 y3 = a1 a2 a3 and y1 + y2 + y3 >= a1 + a2 + a3, y1 y2 + y2 y3 + y1 y3 >= a1 a2 + a2 a3 + a1 a3. Then the following inequality holds (log y1)^2 + (log y2)^2 + (log y3)^2 >= (log a1)^2 + (log a2)^2 + (log a3)^2. This can also be stated in terms of real positive definite 3x3-matrices P1, P2: If their determinants are equal det P1 = det P2, then tr P1 >= tr P2 and tr Cof P1 >= tr Cof P2 implies norm(log P1) >= norm(log P2), where log is the principal matrix logarithm and norm(P) denotes the Frobenius matrix norm. Applications in matrix analysis and nonlinear elasticity are indicated.