The numerical range of a matrix is a set of complex numbers that provides insight into the behavior of the matrix. Specifically, for a given matrix A, the numerical range is defined as the set of values obtained from the inner product ⟨Ax, x⟩, where x is any unit vector in the corresponding vector space. This concept helps in understanding properties like eigenvalues and stability. Numerical ranges are useful in various fields, including quantum mechanics and functional analysis. They can reveal important information about the spectrum of the matrix and assist in solving problems related to operator theory.