On the matrix $A_p$ conjecture

This paper provides a solution to the matrix $A_p$ conjecture. In particular, for every $1<p<\mathrm{\infty }$, we show that the norm of the Hilbert transform on the matrix-weighted space $L^p(W)$ is bounded below by $c_p[W{]}_{A_p}^{1+1\mathrm{/}[p(p-1)]}$ for a positive constant $c_p$ depending only on $p$. This result, combined with the case $p=2$ due to Domelevo, Petermichl, Treil and Volberg \cite{domelevo-petermichl-treil-volberg-2024}, completes the picture of the matrix $A_p$ conjecture.