Appendix A

Chapter Outline

A.1 Properties of Matrices 1013

Matrix inversion lemmas 1014

Matrix derivatives 1014

A.2 Positive Definite and Symmetric Matrices 1015

A.3 Wirtinger Calculus 1016

References 1017

Let A,B,C, and D be matrices of appropriate sizes. Invertibility is always assumed, whenever a matrix inversion is performed. The following properties hold true.

• (AB)^{T} = B^{T}A^{T}.

• (AB)^{−1} = B^{−1}A^{−1}.

• (A^{T})^{−1} = (A^{T})^{−1}.

• trace{AB} = trace{BA}.

• From the previous, we readily get

$trace\{ABC\}=trace\{CAB\}=trace\{BCA\}.$

• det(AB) = det(A)det(B), where det(⋅) denotes the determinant of a square matrix. As ...

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