The Law of Multiplication of Large Random Matrices Revisited
The paper deals with the eigenvalue distribution of the product of two n × n positive definite matrices $B_\tau, \ \tau=\pm 1$, rotated with respect to each other by the random orthogonal and Haar distributed matrix. The problem has been considered in several works by using various techniques. We pr...
Збережено в:
| Дата: | 2023 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна Національної академії наук України
2023
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| Теми: | |
| Онлайн доступ: | https://jmag.ilt.kharkiv.ua/index.php/jmag/article/view/1003 |
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| Назва журналу: | Journal of Mathematical Physics, Analysis, Geometry |
Репозитарії
Journal of Mathematical Physics, Analysis, Geometry| Резюме: | The paper deals with the eigenvalue distribution of the product of two n × n positive definite matrices $B_\tau, \ \tau=\pm 1$, rotated with respect to each other by the random orthogonal and Haar distributed matrix. The problem has been considered in several works by using various techniques. We propose a streamlined approach based on the random matrix theory techniques and a certain symmetry of the problem. We prove the convergence with probability 1 as n tends to infinity of the Normalized Counting Measure (NCM) of eigenvalues of the product to a non-random limit, derive a functional equation that determines the Stieltjes transform of the limiting NCM of the product in terms of limiting NCMs of the factors $B_\tau, \ \tau=\pm 1$, and consider an interesting example.
Mathematical Subject Classification 2020: 15B52, 34L20, 60B20 |
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