余俊伟, 陆明杰, 徐玮玮. 一类酉约束矩阵迹函数最大值问题的解析解[J]. 云南大学学报(自然科学版), 2022, 44(1): 16-22. doi: 10.7540/j.ynu.20210141
引用本文: 余俊伟, 陆明杰, 徐玮玮. 一类酉约束矩阵迹函数最大值问题的解析解[J]. 云南大学学报(自然科学版), 2022, 44(1): 16-22. doi: 10.7540/j.ynu.20210141
YU Jun-wei, LU Ming-jie, XU Wei-wei. Analytical solutions of a class of matrix trace maximization problems with unitary constraints[J]. Journal of Yunnan University: Natural Sciences Edition, 2022, 44(1): 16-22. DOI: 10.7540/j.ynu.20210141
Citation: YU Jun-wei, LU Ming-jie, XU Wei-wei. Analytical solutions of a class of matrix trace maximization problems with unitary constraints[J]. Journal of Yunnan University: Natural Sciences Edition, 2022, 44(1): 16-22. DOI: 10.7540/j.ynu.20210141

一类酉约束矩阵迹函数最大值问题的解析解

Analytical solutions of a class of matrix trace maximization problems with unitary constraints

  • 摘要: 酉约束的矩阵优化问题在数据控制理论和电子结构计算等领域内有着非常广泛的应用. 考虑了一类酉约束的矩阵迹函数最大值问题: \mathop \max \limits_S_kS_k^H = I_n,W_kW_k^H = I_t,V_kV_k^H = I_m \left| \texttr\left( cI_m \pm \prod\limits_k = 1^2 \Gamma _k S_k\Delta _kW_kH_kV_k \right) \right|, 其中 \Gamma _k\Delta _kH_k 分别是 m \times nn \times tt \times m 复对角矩阵,k = 1,2; c 是复数,\texttr( \cdot ) 表示矩阵迹函数,I_mm \times m 单位矩阵. 在前人研究的基础上首先考虑了系数 c 是复数,所研究矩阵是不同维的情况,最后给出数值实验验证了所得结果的有效性.

     

    Abstract: The matrix optimization problem with unitary constraints is widely used in the fields of data control theory and electronic structure calculation. In this paper, we consider a class of unitary constrained matrix trace maximization problems as follows: \mathop \max \limits_S_kS_k^H = I_n,W_kW_k^H = I_t,V_kV_k^H = I_m \left| \texttr\left( cI_m \pm \prod\limits_k = 1^2 \Gamma _k S_k\Delta _kW_kH_kV_k \right) \right|, where \Gamma _k, \Delta _k, H_k are m \times n, n \times t, t \times m complex diagonal matrices respectively, k = 1,2. c \in \mathbfC, \texttr( \cdot ) denotes the matrix trace function, I_m is the m \times m identity matrix. On the basis of predecessor research result, it is considered that the coefficients are complex and the studied matrix is of different dimensions. Finally, numerical experiments are given to verify the effectiveness of the results.

     

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