Non-linear spectra of matrices and extremal problems

Theoretical questions and the computational aspects of the search for the spectral values $\lambda$ and vectors $x\in\mathbb R^m\setminus\{0\}$ of a system $A^{\mathrm T}(Ax)_{(q)}=\lambda^q(x)_{(p)}$, where $A$ is a $k\times m$ matrix, $1\le p$, $q<\infty$ are presented. When $p=q=2$ this is simply the problem of determining the singular values of $A$. Nonlinear systems $((p-2)^2+(q-2)^2\ne0)$ arise in many fields of analysis, mechanics and approximation theory.