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

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.