Matrix Calculus - Derivatives With Matrices

Derivatives With Matrices

There are two types of derivatives with matrices that can be organized into a matrix of the same size. These are the derivative of a matrix by a scalar and the derivative of a scalar by a matrix respectively. These can be useful in minimization problems found many areas of applied mathematics and have adopted the names tangent matrix and gradient matrix respectively after their analogs for vectors.

NOTE: The discussion in this section assumes the numerator layout convention for pedagogical purposes. Some authors use different conventions. The section on layout conventions discusses this issue in greater detail. The identities given further down are presented in forms that can be used in conjunction with all common layout conventions.