Complex-Valued Matrix Derivatives: With Applications in Signal Processing and Communications
In this complete introduction to the theory of finding derivatives of scalar-, vector- and matrix-valued functions with respect to complex matrix variables, Hjørungnes describes an essential set of mathematical tools for solving research problems where unknown parameters are contained in complex-valued matrices. The first book examining complex-valued matrix derivatives from an engineering perspective, it uses numerous practical examples from signal processing and communications to demonstrate how these tools can be used to analyze and optimize the performance of engineering systems. Covering un-patterned and certain patterned matrices, this self-contained and easy-to-follow reference deals with applications in a range of areas including wireless communications, control theory, adaptive filtering, resource management and digital signal processing. Over 80 end-of-chapter exercises are provided, with a complete solutions manual available online.
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2 Background Material
3 Theory of ComplexValued Matrix Derivatives
4 Development of ComplexValued Derivative Formulas
5 Complex Hessian Matrices for Scalar Vector and Matrix Functions
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Assume augmented matrix augmented matrix variable block matrix chain rule chapter CM×P CMxN CMxP CN×N CN×Q CNxl CNxN CNxQ column commutation matrix complex conjugate complex differential complex Hessian matrix Complex-Valued Derivatives complex-valued matrix derivatives complex-valued matrix variable complex-valued vector component function defined Definition denoted derivative of g derivatives with respect diag diffeomorphism duplication matrix dvec dvec(Z dvecT eigenvalue equation Example expressed F with respect find the derivative finding derivatives finding the complex FIR MIMO filter follows formal derivatives function F Hadamard product Hence Hermitian HZ,Zf Hz.zf identified input variables Lemma Let f Magnus and Neudecker main diagonal manifold matrix function Moore-Penrose inverse Neudecker 1988 optimization Palomar parameterization function permutation matrix precoder processing and communications scalar function second-order differential Section Show signal processing Subsection symmetric tangent space Theorem theory unpatterned vecb vecr vector function