Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of t...Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space,but not every infinite matrix corresponds to an operator.The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators,which is considered a respectable mathematical accomplishment.In this paper,we prove the compact version of the Schur test.Moreover,we provide the Schur test for the Schatten class S_(2).It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers.We finally provide the Schur test for compact operators from l_(p) into l_(q).展开更多
本文介绍了相似矩阵、正交矩阵、正定矩阵、幂等矩阵、幂零矩阵、Hermite矩阵的迹的性质及定理,并证明了矩阵的迹的Cauchy-Schwarz公式和Schur公式。并举例说明这些引理和定理在具体解题过程中的应用,充分体现了用矩阵的迹在解决实际问...本文介绍了相似矩阵、正交矩阵、正定矩阵、幂等矩阵、幂零矩阵、Hermite矩阵的迹的性质及定理,并证明了矩阵的迹的Cauchy-Schwarz公式和Schur公式。并举例说明这些引理和定理在具体解题过程中的应用,充分体现了用矩阵的迹在解决实际问题过程中的便捷性的便捷性以及掌握矩阵迹的性质的重要性。This article introduces the properties and theorems of the trace of similarity matrix, orthogonal matrix, positive definite matrix, idempotent matrix, nilpotent matrix, Hermite matrix, and proves the Cauchy Schwarz formula and Schur formula of the trace of matrix. Examples are given to illustrate the application of these property theorems in specific problem-solving processes, fully demonstrating the convenience of using the trace of matrix in solving practical problems and the importance of mastering the properties of matrix trace.展开更多
文摘Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space,but not every infinite matrix corresponds to an operator.The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators,which is considered a respectable mathematical accomplishment.In this paper,we prove the compact version of the Schur test.Moreover,we provide the Schur test for the Schatten class S_(2).It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers.We finally provide the Schur test for compact operators from l_(p) into l_(q).
文摘本文介绍了相似矩阵、正交矩阵、正定矩阵、幂等矩阵、幂零矩阵、Hermite矩阵的迹的性质及定理,并证明了矩阵的迹的Cauchy-Schwarz公式和Schur公式。并举例说明这些引理和定理在具体解题过程中的应用,充分体现了用矩阵的迹在解决实际问题过程中的便捷性的便捷性以及掌握矩阵迹的性质的重要性。This article introduces the properties and theorems of the trace of similarity matrix, orthogonal matrix, positive definite matrix, idempotent matrix, nilpotent matrix, Hermite matrix, and proves the Cauchy Schwarz formula and Schur formula of the trace of matrix. Examples are given to illustrate the application of these property theorems in specific problem-solving processes, fully demonstrating the convenience of using the trace of matrix in solving practical problems and the importance of mastering the properties of matrix trace.
