Linear matrix inequalities (LMIs) are mathematical expressions that involve matrices and linear inequalities. They typically take the form of a matrix inequality, such as A + BXC ≼ D, where X is the variable matrix, and A, B, C, and D are given matrices. LMIs are widely used in control theory, optimization, and systems engineering to express constraints in a structured way. The significance of LMIs lies in their ability to represent complex problems in a linear format, making them easier to solve using numerical methods. They can be applied in various fields, including robotics, finance, and signal processing, to ensure stability and performance in systems.