Linear Matrix Inequalities (LMIs) are mathematical expressions that involve matrices and inequalities. They typically take the form of a linear function of a matrix being less than or equal to another matrix. LMIs are widely used in control theory, optimization, and systems engineering to express constraints and conditions 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 and processes.