Tensor (intrinsic definition)

Orthogonality

Pure mathematics

Bilinear interpolation

Subspace topology

Mathematics

Orthogonalization

Kernel (algebra)

Algorithm

Image (mathematics)

Statistics

Image restoration

Matrix (chemical analysis)

Algebra over a field

Mathematical analysis

Linear algebra

Krylov subspace

Orthogonal complement

Artificial intelligence

Composite material

Numerical linear algebra

Outer product

Materials science

Geometry

Deblurring

Computer science

Tensor product

Applied mathematics

Linear system

Bilinear form

Image processing

SIAM Journal on Matrix Analysis and Applications

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 eigentuples defined in the recent article by Braman. Implications for extending basic algorithms such as the power method, QR iteration, and Krylov subspace methods are discussed. We conclude with two examples in image processing: using the orthogonal elements generated via a Golub--Kahan iterative bidiagonalization scheme for object recognition and solving a regularized image deblurring problem.

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