Hypercubes as operators

Although it is usually easy to determine if a matrix represents a linear operator or a hypercube, there are some corner-cases where a 2-dimensional hypercube is used as a linear operator.

Mostly, the difficulties arise from a variant of the following theorem:

Schmidt-Mirsky Approximation Theorem [19]
Let X=UΣVH be an SVD and define Σk to be Σ where only the largest k elements are non-zero. Then Y=kVH minimizes the difference X-Y, measured in any unitarily invariant norm, for any matrix Y of rank-k.

This theorem is also known as the Eckart-Young theorem, who rediscovered the result (specialized to matrices) about 30 years after Schmidt [18]. Mirsky generalized the theorem to all unitarily invariant norms.

Some variants, more or less equivalent to the theorem above possibly with some preprocessing of the data-matrix, are:

The "problem" with this theorem is that it "allows" one to use the SVD on data-matrices. The easiest way to use BBTools, and remain sane, is to regard a data-matrix used as an operator as a computational trick.


[18] Gilbert Wright Stewart. Perturbation Theory for the Singular Value Decomposition. University of Maryland, 1990. Reference number: CS-TR-2539. (Technical report: )

[19] Gilbert Wright Stewart and Ji-guang Sun. Matrix Perturbation Theory. Academic Press, 1990. ISBN: 0-12-670230-6. (Book)