A definition for functions of multidimensional arrays is presented. The definition is valid for third‐order tensors in the tensor t‐product formalism, which regards third‐order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third‐order tensors and computed for a small‐scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided.

中文翻译：

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张量t函数：三阶张量函数的定义

提出了多维数组功能的定义。该定义对于张量t乘积形式主义中的三阶张量有效，后者将三阶张量视为块循环矩阵。张量函数定义在基本方案中显示出与标准矩阵函数定义类似的属性。为了证明该定义在应用中的潜力，将网络可通信性的概念推广到三阶张量，并通过块Krylov子空间方法对矩阵函数进行小规模示例计算。还提供了在张量上下文中这些方法的复杂度分析。