A representation theorem for (tr \(A^ p)^{1/p}\) (Q1095220)

Let A and B be \(n\times n\) positive semidefinite real matrices, \(p>1\), and let \(q=p/(p-1)\). The main result of the paper is the following quasilinear representation of (tr \(A^ p)^{1/p}:\) \(tr(A^ p)^{1/p}=_{tr B^ q=1}tr(AB).\) Using this it is shown that \(tr(AB)\leq (tr A^ p)^{1/p}(tr B^ q)^{1/q}\) and \((tr(A+B)^ p)^{1/p}\leq (tr A^ p)^{1/p}+(tr B^ p)^{1/p}\) which are matrix analogues of the Hölder and Minkowski inequalities, respectively.