A third order real tensor is mapped to a special f-diagonal tensor by going through Discrete Fourier Transform (DFT), standard matrix SVD and inverse DFT. We call such an f-diagonal tensor an s-diagonal tensor. An f-diagonal tensor is an s-diagonal tensor if and only if it is mapped to itself in the above process. The third order tensor space is partitioned to orthogonal equivalence classes. Each orthogonal equivalence class has a unique s-diagonal tensor. Two s-diagonal tensors are equal if they are orthogonally equivalent. Third order tensors in an orthogonal equivalence class have the same tensor tubal rank and T-singular values. Four meaningful necessary conditions for s-diagonal tensors are presented. Then we present a set of sufficient and necessary conditions for s-diagonal tensors. Such conditions involve a special complex number. In the cases that the dimension of the third mode is $2, 3$ and $4$, we present direct sufficient and necessary conditions which do not involve such a complex number.
翻译：将第三顺序的真抗拉映射到特殊的 f- 直角变形( DFT)、 标准矩阵 SVD 和反 DFT 。 我们称此 f- 直角振动为 S- 直角振动。 f- 直角振动是 s- 直角振动, 仅在上述进程中将它映射到自己的位置。 第三顺序的振动空间被分割为正对等等级。 每个正对等级都有独特的 s- 直角阵列。 两个 s- 直角阵列是等的。 两个 s- 直角阵列是等同的。 一个或正对等型的第三顺序是 10, 具有相同的色调管级和 T- 等值。 显示了 s- 直角阵列阵列的四种有意义的必要条件 。 然后我们为 s- diagonal 发动阵列提供了一套足够和必要的条件 。 这些条件涉及一个特殊的复杂数目。 在第三个模式的维度为 2, 3 和 4$ 的情况下, 我们展示了足够复杂和必要的条件 。
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