Elliptic system of differential equations ($\det A_N(\xi )\ne 0(\xi \ne 0)$)

A square system whose entries have common maximal order $N$ is elliptic when its matrix principal symbol is invertible at every nonzero real frequency. Unlike scalar operators, a system can be first-order elliptic in three dimensions: the Pauli matrices give $A(\xi )={\sum }_j{\sigma }_j{\xi }_j$ with $A(\xi {)}^2=|\xi {|}^{2}1$.