Trace Properties

For the square matric A, define tr A = trace(A) = $\sum_{i}A_{i,i}$. Let $\bigtriangledown f(A)$ be the matrix of partial derivatives of f with respect to the elements of A. The following hold:

  1. tr AB = tr BA
  2. $\bigtriangledown tr AB = B^T$
  3. $tr A = tr A^T$
  4. $\bigtriangledown tr ABA^TC = CAB + C^TAB^T$

Transpose Properties

  1. $(A+B)^T= A^T+ B^T$
  2. $(AB)^T = B^TA^T$
  3. $det(A^T) = det(A)$
  4. $(A^T)^{-1}=(A^{-1})^T$

Inverse Properpties

  1. $(AB)^{-1} = B^{-1}A^{-1}$

Matrix Derivatives

  1. $\frac{\partial}{\partial \mathbf{x}} (\mathbf{x}^T \mathbf{y}) = \frac{\partial}{\partial \mathbf{x}} (\mathbf{y}^T \mathbf{x}) = \mathbf{y}$
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