Complex-Valued Matrix Derivatives: With Applications in by Are Hjørungnes

By Are Hjørungnes

During this entire advent to the idea of discovering derivatives of scalar-, vector- and matrix-valued features with recognize to complicated matrix variables, Hjørungnes describes an important set of mathematical instruments for fixing study difficulties the place unknown parameters are contained in complex-valued matrices. the 1st publication studying complex-valued matrix derivatives from an engineering viewpoint, it makes use of quite a few sensible examples from sign processing and communications to illustrate how those instruments can be utilized to research and optimize the functionality of engineering structures. overlaying un-patterned and sure patterned matrices, this self-contained and easy-to-follow reference offers with purposes in quite a number parts together with instant communications, keep an eye on conception, adaptive filtering, source administration and electronic sign processing. Over eighty end-of-chapter routines are supplied, with an entire recommendations guide on hand on-line.

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14) respectively. 12) as two linear equations with the two unknowns ∂∂x f (z 0 ) and ∂∂y f (z 0 ). 2 can be extended. 2 can be identified from the complex differentials of these functions. 17) 1 ∂z ∗ = ∂z 2 ∂ ∂ − ∂x ∂y (x −  y) = 1 (1 − 1) = 0. 1), only derivatives ∗ = 1 but dz does not exist. 2 Let the function f : C × C → R given by f (z, z ∗ ) = zz ∗ . 19) f (z, z ∗ ) = z ∗ , ∂z ∂ f (z, z ∗ ) = z. 20) ∂z ∗ When the complex variable z and its complex conjugate twin z ∗ are treated as independent variables (Brandwood 1983, Theorem 1), then the function f is differentiable in both of these variables.

4 Let A ∈ C N ×M and B ∈ C M×N , then det (I N + AB) = det (I M + B A) . 68) IN, A . 65). 5 Let A ∈ C N ×M and B ∈ C M×N . The N × N matrix I N + AB is invertible if and only if the M × M matrix I M + B A is invertible. If these two matrices are invertible, then B (I N + AB)−1 = (I M + B A)−1 B. 67), it follows that I N + AB is invertible if and only if I M + B A is invertible. By multiplying out both sides, it can be seen that the following relation holds: B (I N + AB) = (I M + B A) B. 69). 4).

N − 1}. 5 Useful Manipulation Formulas In this section, several useful manipulation formulas are presented. Although many of these results are well known in the literature, they are included here to make the text more complete. A classical result from linear algebra is that if A ∈ C N ×Q , then (Horn & Johnson 1985, p. 13) rank (A) + dimC (N (A)) = Q. 55) The following lemma states Hadamard’s inequality (Magnus & Neudecker 1988), and it will be used in Chapter 6 to derive the water-filling solution of the capacity of MIMO channels.

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