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

作者：Birsan Mircea; Neff Patrizio^*; Lankeit Johannes

来源：Journal of Inequalities and Applications, 2013, 2013(1): 168.

DOI：10.1186/1029-242X-2013-168

• matrix logarithm
• elementary symmetric polynomials
• inequality
• characteristic polynomial
• positive definite matrices
• means

Let y(1), y(2), y(3), a(1), a(2), a(3) is an element of (0,infinity) be such that y(1)y(2)y(3) = a(1)a(2)a(3) and %26lt;br%26gt;y(1) + y(2) + y(3) %26gt;= a(1) + a(2) + a(3), y(1)y(2) + y(2)y(3) + y(1)y(3) %26gt;= a(1)a(2) + a(2)a(3) + a(1)a(3). %26lt;br%26gt;Then %26lt;br%26gt;(log y(1))(2) + (log y(2))(2) + (log y(3))(2) %26gt;= (log a(1))(2) + (log a(2))(2) + (log a(3))(2). %26lt;br%26gt;This can also be stated in terms of real positive definite 3 x 3-matrices P-1, P-2: If their determinants are equal, det P-1 = det P-2, then %26lt;br%26gt;tr P-1 %26gt;= tr P-2 and tr Cof P-1 %26gt;= tr Cof P-2 double right arrow parallel to log P-1 parallel to(2)(F) %26gt;= parallel to log P-2 parallel to(2)(F), %26lt;br%26gt;where log is the principal matrix logarithm and parallel to P parallel to(2)(F) = Sigma(3)(i,j=1) P-ij(2) denotes the Frobenius matrix norm. Applications in matrix analysis and nonlinear elasticity are indicated.

• 出版日期2013