Spectral theory is a branch of mathematics that studies the spectrum of operators, particularly in the context of linear algebra and functional analysis. It focuses on understanding how linear transformations can be represented in terms of their eigenvalues and eigenvectors, which are fundamental concepts that describe how a system behaves under certain conditions. In physics and engineering, spectral theory is essential for analyzing systems such as quantum mechanics and vibrations. It helps in solving differential equations and understanding stability, resonance, and other phenomena by examining the properties of matrices and linear operators in various spaces.