Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Every complex Hermitian tensor is a sum of complex Hermitian rank-1 tensors. However, this is not true for the real case. We study basic properties for Hermitian tensors such as Hermitian decompositions and Hermitian ranks. For canonical basis tensors, we determine their Hermitian ranks and decompositions. For real Hermitian tensors, we give a full characterization for them to have Hermitian decompositions over the real field. In addition to traditional flattening, Hermitian tensors specially have Hermitian and Kronecker flattenings, which may give different lower bounds for Hermitian ranks. We also study other topics such as eigenvalues, positive semidefiniteness, sum of squares representations, and separability.

中文翻译：

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厄米张量分解

Hermitian 张量是 Hermitian 矩阵的推广，但它们具有非常不同的性质。每个复 Hermitian 张量都是复 Hermitian rank-1 张量的总和。然而，对于实际情况，情况并非如此。我们研究 Hermitian 张量的基本属性，例如 Hermitian 分解和 Hermitian 秩。对于规范基张量，我们确定它们的 Hermitian 等级和分解。对于真正的 Hermitian 张量，我们给出了它们的完整表征，以便在实场上进行 Hermitian 分解。除了传统的扁平化之外，Hermitian 张量还特别有 Hermitian 和 Kronecker 扁平化，这可能会给 Hermitian 秩给出不同的下界。我们还研究了其他主题，例如特征值、半正定性、平方和表示和可分离性。