The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the orbits of ${\mathrm{GU}}_m\left(q\right)\times {\mathrm{GU}}_n\left(q\right)$ on $M_{m\times n}\left(q^2\right)$ (which is the analog of the singular value decomposition). The proof involves Kronecker’s theory of pencils and the Lang–Steinberg theorem for algebraic groups. Besides the motivation mentioned above, this problem came up in a recent paper of Guralnick et al. (2020) where a concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups was studied and bounds on the number of orbits was needed. A consequence of this work determines possible pairs of Jordan forms for nilpotent matrices of the form $A{A}^{\ast }$ and $A^{\ast }A$ over a finite field and $A{A}^{\top }$ and $A^{\top }A$ over arbitrary fields.

复矩阵的奇异值分解是线性代数中的一个基本概念，并且已被证明在许多学科中都非常有用。不太清楚有限域上的情况。在本文中，我们对轨道进行分类${\mathrm{顾}}_{米}\left(q\right)\times {\mathrm{顾}}_n\left(q\right)$ 在 ${米}_{米\times n}\left(q^2\right)$（这是奇异值分解的类比）。证明涉及 Kronecker 的铅笔理论和代数群的 Lang-Steinberg 定理。除了上述动机之外，这个问题出现在 Guralnick 等人最近的一篇论文中。(2020) 研究了有限群、一般群或特殊群、线性群和酉群的复杂不可约特征的特征水平概念，并需要轨道数的界限。这项工作的结果确定了形式的幂零矩阵的可能的约旦形式对$\mathrm{一种}{\mathrm{一种}}^{＊}$ 和 ${\mathrm{一种}}^{＊}\mathrm{一种}$ 在有限域上和 $\mathrm{一种}{\mathrm{一种}}^{\top }$ 和 ${\mathrm{一种}}^{\top }\mathrm{一种}$ 在任意领域。