Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4601121 | Linear Algebra and its Applications | 2012 | 6 Pages |
Abstract
A capital letter means n×nn×n matrix. T is said to be positive definite (denoted by T>0T>0) if T is positive semidefinite and invertible. We shall show the following central results via log majorization obtained by an order preserving operator inequality.Theorem.If A>0A>0and B⩾0B⩾0, then for 0⩽α⩽1,t∈[0,1]0⩽α⩽1,t∈[0,1]and r⩾tr⩾tA1-t2At♯αBA1-t2s≻(log)w2(Ar♯αBs)Aw2holds for (1-α)(r-t)1-αt+1⩾s⩾1,where w=(1-α)(s-r)+α(1-t)sw=(1-α)(s-r)+α(1-t)s.Our result extends the following recent elegant inequality by Matharu and Aujlia.Let A,BA,B be positive definite and α∈[0,1].α∈[0,1]. Then∏i=1kλj(A1-αBα)⩾∏i=1kλj(A♯αB)1⩽k⩽n.Also some results associated with log majorization are shown.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Takayuki Furuta,