It was shown recently that the f-diagonal tensor in the T-SVD factorization must satisfy some special properties. Such f-diagonal tensors are called s-diagonal tensors. In this paper, we show that such a discussion can be extended to any real invertible linear transformation. We show that two Eckart-Young like theorems hold for a third order real tensor, under any doubly real-preserving unitary transformation. The normalized Discrete Fourier Transformation (DFT) matrix, an arbitrary orthogonal matrix, the product of the normalized DFT matrix and an arbitrary orthogonal matrix are examples of doubly real-preserving unitary transformations. We use tubal matrices as a tool for our study. We feel that the tubal matrix language makes this approach more natural.

翻译：最近显示, T- SVD 系数化中的 F- diagonal 发光器必须满足某些特殊特性。 这种 F- diagonal 发光器被称为 s- diagonal 发光器。 在本文中, 我们显示, 这样的讨论可以扩展到任何真实的不可逆线性变异。 我们显示, 两个Eckart- Youngng像理论体一样, 在任何双重的、真实的、 保存的单一变异器下, 维持第三顺序的真发光器 。 常规的 Discrete Fourier 变异矩阵、 一个任意的正方形矩阵、 常规的 DFT 矩阵的产物以及任意的正方形变光器是双重的、 真实的单一变异的示例。 我们用管矩阵作为研究的工具。 我们认为, 管式矩阵语言使得这一方法更加自然。