Neutrinos stand out among the elementary particles because of their unusually small masses. Various seesaw mechanisms attempt to explain this fact. In this work, applying insights from matrix theory, we are in a position to treat variants of seesaw mechanisms in a general manner. Specifically, using Weyl's inequalities, we discuss and rigorously prove under which conditions the seesaw framework leads to a mass spectrum with exactly three light neutrinos. We find an estimate of the mass of heavy neutrinos to be the mass obtained by neglecting light neutrinos, shifted at most by the maximal strength of the coupling to the light neutrino sector. We provide analytical conditions allowing one to prescribe that precisely two out of five neutrinos are heavy. For higher-dimensional cases the inverse eigenvalue methods are used. In particular, for the CP-invariant scenarios we show that if the neutrino sector has a valid mass matrix after neglecting the light ones, i.e. if the respective mass submatrix is positive definite, then large masses are provided by matrices with large elements accumulated on the diagonal. Finally, the Davis-Kahan theorem is used to show how masses affect the rotation of light neutrino eigenvectors from the standard Euclidean basis. This general observation concerning neutrino mixing, together with results on the mass spectrum properties, opens directions for further neutrino physics studies using matrix analysis.

中微子因其异常小的质量而在基本粒子中脱颖而出。各种跷跷板机制试图解释这一事实。在这项工作中，应用矩阵理论的见解，我们能够以一般方式处理跷跷板机制的变体。具体来说，我们使用外尔不等式，讨论并严格证明在什么条件下跷跷板框架会导致质谱恰好具有三个轻中微子。我们发现对重中微子质量的估计是通过忽略轻中微子获得的质量，最多偏移到轻中微子扇区的耦合的最大强度。我们提供的分析条件允许人们精确地规定五分之二的中微子是重的。对于更高维的情况，使用反特征值方法。特别是，对于CP不变场景我们表明，如果中微子扇区在忽略轻质子后具有有效的质量矩阵，即如果相应的质量子矩阵是正定的，则​​大质量由对角线上累积的大元素的矩阵提供。最后，Davis-Kahan 定理用于说明质量如何影响标准欧几里得基中轻中微子特征向量的旋转。这种关于中微子混合的一般性观察，连同质谱特性的结果，为使用基质分析进一步研究中微子物理开辟了方向。