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

By Are Hjørungnes

During this whole advent to the speculation of discovering derivatives of scalar-, vector- and matrix-valued services with recognize to advanced matrix variables, Hjørungnes describes an important set of mathematical instruments for fixing learn difficulties the place unknown parameters are contained in complex-valued matrices. the 1st booklet interpreting complex-valued matrix derivatives from an engineering point of view, it makes use of various sensible examples from sign processing and communications to illustrate how those instruments can be utilized to research and optimize the functionality of engineering platforms. overlaying un-patterned and likely patterned matrices, this self-contained and easy-to-follow reference bargains with functions in 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 workouts are supplied, with a whole ideas handbook to be had on-line.

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**Example text**

The duplication matrix D N has size N 2 × N (N2+1) and is left invertible by its MoorePenrose inverse, which is given by Magnus and Neudecker (1988, p. 49): D +N = D TN D N −1 D TN . 48) by D +N , the following relation holds: v (A) = D +N vec (A) . 177) by D +N , respectively. 185) where z = x + y, Re{ f (z, z ∗ )} = u(x, y), and Im{ f (z, z ∗ )} = v(x, y). 186) ∂u ∂v =− . 2 Functions that are going to be maximized or minimized must be real-valued. The results of this exercise show that in engineering problems of practical interests, the objective functions that are interesting do not satisfy the Cauchy-Riemann equations.

15 (Matrices V d , V l , and V) Let A ∈ C N ×N be symmetric. Unique matriN (N +1) ×N N (N +1) × N (N −1) 2 ces V d ∈ Z2 2 and V l ∈ Z2 2 contain zeros everywhere except for +1 at one place in each column, and these matrices can be used to build up v(A), from vecd (A) and vecl (A) in the following way: v (A) = V d vecd (A) + V l vecl (A) = [V d , V l ] N (N +1) × N (N2+1) 2 The square permutation matrix V ∈ Z2 vecd (A) vecl (A) . 49) is defined by V = [V d , V l ] . 52) V dT V l = 0 N × (N −1)N .

10 (Diagonalization Operator) Let a ∈ C N ×1 , and let the i-th vector component of a be denoted by ai , where i ∈ {0, 1, . . , N − 1}. 4 Matrix-Related Definitions vecd (A) a 0,0 a1,0 a2,0 .. . aN −1,0 vecu (A) ··· a0,1 a0,2 a1,1 a1,2 a1,3 a2,1 a2,2 a2,3 a3,1 .. a3,2 .. a0,N −1 a1,N −1 a ··· ··· ... 1 The way the three operators vecd (·), vecl (·), and vecu (·) are returning their elements from the matrix A ∈ C N ×N . The operator vecd (·) returns the elements on the line along the main diagonal, starting in the upper left corner and going down along the main diagonal; the operator vecl (·) returns elements along the curve below the main diagonal following the order indicated in the figure; and the operator vecu (·) returns elements along the curve above the main diagonal in the order indicated by the arrows along that curve.