Modification in Functions of Scalar and Matrix Argument: Defining the Gradient for Matrix Functionals with Symmetric Arguments (Journal of Advances and Scholarly Researches in Allied Education, 2020)
Matrix functional defined over an inner-product space of square matrices are a common construct in applied mathematics. In most cases, the object of interest is not the matrix functional itself, but its derivative or gradient (if it be differentiable), and this notion is unambiguous. The Frechet derivative, see for e.g. and, being a linear functional readily yields the definition of the gradient via the Riesz Representation Theorem. However, there is a sub-class of matrix functional that frequently occurs in practice whose argument is a symmetric matrix. For instance, in the theory of elasticity and continuum thermodynamics, the stress (a second-order, symmetric tensor) is defined to be the gradient of the strain energy functional or Helmholtz potential with respect to the (symmetric) strain tensor while the strain is defined to be the gradient of the Gibbs potential with respect to the stress. Such functional and their gradients also occur in the analysis and control of dynamical systems, which are described by matrix differential equations, and maximum likelihood estimation in statistics, econometrics and machine-learning. For this sub-class of matrix functional with symmetric arguments, there seem to be two approaches to define the gradient that lead to different results.