How to find derivative of squared Frobenius Norm of XX transpose

Question

Can anyone show me how to get the derivative of the squared Frobenius Norm for this second degree matrix $X$?

$$||XX^T||_F^2$$

Answer 1

Use a colon to denote the matrix inner product.

Write down the function and calculate its gradient as follows $$\eqalign{ \phi &= \big\|XX^T\big\|_F^2 \\ &= XX^T:XX^T \\ d\phi &= 2XX^T:d(XX^T) \\ &= 2XX^T:\Big(X\,dX^T+dX\,X^T\Big) \\ &= 2\Big(XX^T+(XX^T)^T\Big):dX\,X^T \\ &= 4XX^TX:dX \\ \frac{\partial \phi}{\partial X} &= 4XX^TX \\ }$$ This is analogous to the scalar $({\tt1}\times{\tt1})$ derivative result $$\frac{d X^4}{dX}=4X^3$$