Some inequalities of majorization type (Q442657)

Some majorization inequalities on real vectors are provided and applied to derive some inequalities concerning norm, eigenvalues, singular values and traces of matrices.NEWLINENEWLINEFor a vector \(x=(x_1,x_2,\dots,x_n)\in{\mathbb R}^n\) one denotes by \(x^{\downarrow}=(x^{\downarrow}_1,x^{\downarrow}_2,\dots,x^{\downarrow}_n)\) the vector having the components of \(x\) rearranged in decreasing order.NEWLINENEWLINEFor \(x=(x_1,x_2,\dots,x_n)\) and \(y=(y_1,y_2,\dots,y_n)\) one says that \(x\) is weakly majorized by \(y\) and one denotes \(x\prec_wy\) if \(\sum_{i=1}^kx^{\downarrow}_i\leq\sum_{i=1}^ky^{\downarrow}_i\), \(\forall k=1,2,\dots,n\).NEWLINENEWLINEIf \(x\) is weakly majorized by \(y\) and \(\sum_{i=1}^kx^{\downarrow}_i=\sum_{i=1}^ky^{\downarrow}_i\) then one says that \(x\) is majorized by \(y\) and one denotes \(x\prec y\). By replacing the sums \(\sum\) by the products \(\prod\) one obtains the definitions of \(x\prec_{wlog}y\) and \(x\prec_{log}y\) (respectively \(x\) is weakly log majorized by \(y\) and \(x\) is log majorized by \(y\).NEWLINENEWLINESome majorization inequalities for vectors are provided, for instance {\parindent=6mm \begin{itemize}\item[i)] \(\frac{1}{m}(x_1,x_2,\dots,x_m)\prec(r_1y_1,r_2y_2,\dots,r_my_m)\) if \(x_i,y_i\in{\mathbb R}^n\) and \(x_i\prec r_1y_1+r_2y_2+\cdots+r_my_m\), \(i=1,2,\dots m\). \item[ii)] \(\frac{1}{m}(x_1,x_2,\dots,x_m)\prec_w(r_1y_1,r_2y_2,\dots,r_my_m)\) if \(x_i,y_i\in{\mathbb R}^n_+\) and \(x_i\prec_wy_1^{r_1}\circ y_2^{r_2}\circ\cdots\circ y_m^{r_m}\), \(i=1,2,\dots m\), where \(\circ\) denotes the componentwise product of two vectors. NEWLINENEWLINE\end{itemize}} The special case \(m=2\) is applied to generate many matrix inequalities, for example:NEWLINENEWLINE\((\sigma(\frac{A+B}{2}),(\sigma(\frac{A+B}{2}))\prec_w(\sigma(A),(\sigma(B))\),NEWLINENEWLINE\((\sigma(AB),(\sigma(AB))\prec_{log}(\sigma^2(A),(\sigma^2(B))\),NEWLINENEWLINE\(\frac{1}{2}\|(A+B)\oplus(A+B)\|\leq\|A\oplus B\|\),NEWLINENEWLINE\(\|(AB)\oplus(AB)\|\leq\|A^*A\oplus B^*B\|\),NEWLINENEWLINE\(\|(A\circ B)\oplus(A\circ B)\|\leq\|A^*A\oplus B^*B\|\),NEWLINENEWLINEwhere for \(A\in M_n\), \(\sigma(A)\) denotes the singular value vector of A with the singular values arranged in decreasing order.NEWLINENEWLINEA generalization and a different proof is provided for a recent result on majorization, concerning partitioned Hermitian positive semidefinite matrices.