Computation of Outer Inverse of Tensors Based on t‐Product

Abstract

Tensor computations play an essential role in various fields of science and engineering, including multiway data analysis. In this study, we established a few basic properties of the range and null space of a tensor by using block circulant matrices and a discrete Fourier matrix. We then discuss the outer inverse of the tensors based on ‐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 ‐QR decomposition based algorithm was developed to compute outer inverses. It is well known that the confidentiality of information transmitted through the virtual world grows exponentially, and color image and video security have become a significant concern when communicating over the internet. As an application, a ‐QR decomposition based algorithm was demonstrated for concealing secret color images and videos.