# What's the more efficient way to solve this matrix equation?

This is intended to be a more generic question not about a specific system. Given a hermitian matrix $$H(x_1,\dots,x_n)$$ depending non-linearly on some real parameters $$x_1,\dots,x_n$$. We want these to be determined numerically such that $$H=0$$ and assume we already know that a set of parameters always exists such that this holds.

Would it be more efficient to directly solve the above equation or to optimise its Frobenius norm $$||H||_F^2$$ (which would just be a scalar)? Also, which method would you apply?

• Could you clarify what you're solving for? It looks as though you're seeking the zero matrix. – Richard Jul 25 '19 at 14:49
• @Richard Indeed, solve for $x_1,\dots,x_n$ such that $H$ vanishes. – user32355 Jul 25 '19 at 15:40
• How is that different from setting all of $x_1,\ldots,x_n$ to zero? – Richard Jul 25 '19 at 15:48
• It really doesn't matter that you have a matrix -- you could just think of a matrix as a vector with funnily-arranged entries. So this is no different to solving the vector equation $\vec F(\vec x)=0$, which is always done using a Newton method (or variants thereof). – Wolfgang Bangerth Jul 25 '19 at 15:56
• Optimize the Frobenius norm might be a bad idea since the objective function can have local extrema that are different from zero. – nicoguaro Jul 25 '19 at 16:47