Tensor operations play an essential role in various fields of science and engineering, including multiway data analysis. In this study, we establish a few basic properties of the range and null space of a tensor using block circulant matrices and the discrete Fourier matrix. We then discuss the outer inverse of tensors based on $t$-product with a prescribed range and kernel of third-order tensors. We address the relation of this outer inverse with other generalized inverses, such as the Moore-Penrose inverse, group inverse, and Drazin inverse. In addition, we present a few algorithms for computing the outer inverses of the tensors. In particular, a $t$-QR decomposition based algorithm is developed for computing the outer inverses.

翻译：张量运算在包括多路数据分析在内的科学与工程各领域中发挥着重要作用。本研究利用分块循环矩阵和离散傅里叶矩阵，建立了张量值域与零空间的一些基本性质。随后，我们探讨了基于$t$-乘积、具有给定值域和核的三阶张量的外逆问题。阐述了该外逆与Moore-Penrose逆、群逆和Drazin逆等其他广义逆之间的关系。此外，我们提出了几种计算张量外逆的算法。特别地，开发了一种基于$t$-QR分解的算法用于计算外逆。