Let A_i, P, i = 1, …, m, be n by n complex matrices such that each A_i is positive definite. It is shown, among other inequalities, that

1. If 0 < a_j ≤ s_n(A_j), $C_m=\sum _{j=1}^m(\sum _{i=1,i\ne j}^m{A}_i{)}^{a_j}$ and P is Hermitian, then for every unitarily invariant norm $\parallel |.\parallel |$, $\parallel |C_{m}P+P{C}_m\parallel |\ge \alpha \parallel |P\parallel |,$ where $\alpha =max\left\{2\left(m-1\right),m\left(1+min\left\{a_1^2,\dots ,a_m^2\right\}\right)\right\}.$

2. If s₁(A_j) ≤ b_j, $\sum _{i=1}^m{b}_i=1$ and $D_m=\sum _{j=1}^m(\sum _{i=1,i\ne j}^m{A}_i{)}^{b_j}$, then for every unitarily invariant norm $\parallel |.\parallel |$, $2m\left|\left|\left|P\right|\right|\right|\ge \parallel |D_{m}P+P{D}_m\parallel |.$

中文翻译：

---

正定矩阵某些幂的进一步不等式

令A _i , P , i  = 1, …, m为n × n 个复矩阵，使得每个A _i均为正定矩阵。它表明，除其他不等式外，

1. 如果 0 <  a _j  ≤  s _n ( A _j )，$C_{米}=\sum _{j=\mathrm{1个}}^{米}(\sum _{我=\mathrm{1个},我\ne j}^{米}{A}_{我}{)}^{A_j}$并且P是 Hermitian，那么对于每个酉不变范数$\parallel |.\parallel |$,$\parallel |C_{米}P+P{C}_{米}\parallel |\ge \alpha \parallel |P\parallel |,$在哪里$\alpha =最大限度\left\{\mathrm{2个}\left(米-\mathrm{1个}\right),米\left(\mathrm{1个}+分钟\left\{A_{\mathrm{1个}}^{\mathrm{2个}},\dots \dots ,A_{米}^{\mathrm{2个}}\right\}\right)\right\}.$

2. 如果s ₁ ( A _j ) ≤  b _j，$\sum _{我=\mathrm{1个}}^{米}{b}_{我}=\mathrm{1个}$和${丁}_{米}=\sum _{j=\mathrm{1个}}^{米}(\sum _{我=\mathrm{1个},我\ne j}^{米}{A}_{我}{)}^{b_j}$, 然后对于每个单一不变范数$\parallel |.\parallel |$,$\mathrm{2个}米\left|\left|\left|P\right|\right|\right|\ge \parallel |{丁}_{米}P+P{丁}_{米}\parallel |.$