Matrix Calculus
Posted on Sat 20 October 2018 in Math
Matrix Calculus
Vector-by-Vector
- \(\frac{\partial a}{\partial x} = 0\)
- \(\frac{\partial x}{\partial x} = I\)
- \(\frac{\partial x}{\partial x} = A\)
- \(\frac{\partial x^TA}{\partial x} = A^T\)
- \(\frac{\partial au(x)}{\partial x} = a\frac{\partial u}{\partial x}\)
Scalar-by-Vector
- \(\frac{\partial a^Tx}{\partial x} = a^T\)
- \(\frac{\partial u^Tv}{\partial x} = u^T\frac{\partial v}{\partial x} + v^T\frac{\partial u}{\partial x}\)
- \(\frac{\partial u^TAv}{\partial x} = u^TA\frac{\partial v}{\partial x} + v^TA^T\frac{\partial u}{\partial x}\)
- \(\frac{\partial x^TAx}{\partial x} = x^T(A^T + A)\)