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Appendix A

# Linear Algebra

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

## A.1 Properties of Matrices

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 = BTAT.

(AB)−1 = B−1A−1.

(AT)−1 = (AT)−1.

trace{AB} = trace{BA}.

From the previous, we readily get

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

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

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