Third-Order Tensors as Operators on Matrices: A Theoretical and Computational Framework with Applications in Imaging

Recent work by Kilmer and Martin [Linear Algebra Appl., 435 (2011), pp. 641--658] and Braman [Linear Algebra Appl., 433 (2010), pp. 1241--1253] provides a setting in which the familiar tools of linear algebra can be extended to better understand third-order tensors. Continuing along this vein, this paper investigates further implications including (1) a bilinear operator on the matrices which is nearly an inner product and which leads to definitions for length of matrices, angle between two matrices, and orthogonality of matrices, and (2) the use of t-linear combinations to characterize the range and kernel of a mapping defined by a third-order tensor and the t-product and the quantification of the dimensions of those sets. These theoretical results lead to the study of orthogonal projections as well as an effective Gram--Schmidt process for producing an orthogonal basis of matrices. The theoretical framework also leads us to consider the notion of tensor polynomials and their relation to tensor eigentupl...

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