Theory of Complex-Valued Matrix Derivatives
Are Hjørungnes · Cambridge University Press eBooks · 2011
Introduction A theory developed for finding derivatives with respect to real-valued matrices with independent elements was presented in Magnus and Neudecker (1988) for scalar, vector, and matrix functions. There, the matrix derivatives with respect to a real-valued matrix variable are found by means of the differential of the function. This theory is extended in this chapter to the case where the function depends on a complex-valued matrix variable and its complex conjugate, when all the elements of the matrix are independent. It will be shown how the complex differential of the function can be used to identify the derivative of the function with respect to both the complex-valued input matrix variable and its complex conjugate. This is a natural extension of the real-valued vector derivatives in Kreutz-Delgado (2008) and the real-valued matrix derivatives in Magnus and Neudecker (1988) to the case of complex-valued matrix derivatives. The complex-valued input variable and its complex conjugate should be treated as independent when finding complex matrix derivatives. For scalar complex-valued functions that depend on a complex-valued vector and its complex conjugate, a theory for finding derivatives with respect to complex-valued vectors, when all the vector components are independent, was given in Brandwood (1983). This was extended to a systematic and simple way of finding derivatives of scalar, vector, and matrix functions with respect to complex-valued matrices when the matrix elements are independent (Hjørungnes & Gesbert 2007 a ). In this chapter, the definition of the complex-valued matrix derivative will be given, and a procedure will be presented for how to obtain the complex-valued matrix derivative.