Fu\v{c}\'{\i}k spectrum for discrete systems: curves and their tangent lines

In this paper, we study the Fu\v{c}\'{\i}k spectrum of a square matrix $A$ and provide necessary and sufficient conditions for the existence of Fu\v{c}\'{\i}k curves emanating from the point $(\lambda,\lambda)$ with $\lambda$ being a real eigenvalue of $A$. We extend recent results by Maroncelli (2024) and remove his assumptions on symmetry of $A$ and simplicity of $\lambda$. We show that the number of Fu\v{c}\'{\i}k curves can significantly exceed the multiplicity of $\lambda$ and determine all the possible directions they can emanate in. We also treat the situation when the algebraic multiplicity of $\lambda$ is greater than the geometric one and show that in such a case the Fu\v{c}\'{\i}k curves can loose their smoothness and provide the slopes of their "one-sided tangent lines". Finally, we offer two possible generalizations: the situation off the diagonal and Fu\v{c}\'{\i}k spectrum of a general Fredholm operator on the Hilbert space with a lattice structure.