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 Discrete Fourier Transformation matrix and any orthogonal matrices are doubly real-preserving unitary transformations. In particular, Discrete Cosine Transformation is in this category. 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 shronor 必须满足某些特殊特性。 这种 f- diagonal Excessors 被称为 s- diagonal Excrons 。 在本文中, 我们显示, 这种讨论可以扩展到任何真实的不可逆线性转换。 我们显示, 两个Eckart- Youngng像理论体一样, 在任何双重的、 真实的、 保存单一的变换下, 持有第三顺序的真电压。 分解器Fourier 变换矩阵和任何正方位矩阵是双重的、 真正保存单一的变换。 特别是, Discrete Cosine 变换属于这个类别。 我们使用管状矩阵作为研究的工具。 我们认为, 管状矩阵语言使得这个方法更加自然。
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