Title: A unified treatment of matrix calculus on structured spaces

The modern viewpoint of calculus and linear algebra focusses on a study of derivatives of functions between normed linear spaces or inner-product spaces, a unified treatment that encompasses calculus of vector and matrix spaces.
However, the practice of matrix calculus as seen in most published work seems to follow an approach antithetical to this modern viewpoint.
As a result, there is no established technique to calculate the gradient for scalar functions defined on \emph{any} arbitrary matrix subspace.
Moreover, the gradient derived from extant methods is incorrect for functions defined on some special matrix subspaces (such as spaces of symmetric or skew-symmetric matrices), and these results populate several publications, as well as respected textbooks and handbooks on matrix calculus.

One of our important contributions has been to wade through the published works carrying these wrong results and set up the calculation of the gradient defined on matrix subspaces in a rigorous and concrete mathematical setting of a finite-dimensional inner-product space.
Following the reasoning behind the spurious results in matrix calculus, we cast the derivative calculation in terms of a scalar function of a vector and in this process, uncover the reason behind the spurious results obtained for various matrix subspaces.
We also propose an alternative to correct the fundamental flaw.

Short Bio

Shriram Srinivasan earned his M.S. in Mathematics and Ph.D. in Mechanical Engineering, all from Texas A & M University, College Station. His research interests are in computational mechanics and reduced-order models of structured systems such as fracture networks and gas pipeline networks. He is a staff scientist in the Applied Mathematics and Plasma Physics Group of the Theoretical Division at Los Alamos National Laboratory, New Mexico.