Matrix inequalities are mathematical expressions that involve matrices and specify a relationship between them, often in terms of positive definiteness or semi-definiteness. For example, a matrix A is said to be positive definite if, for any non-zero vector x, the expression x^T A x is greater than zero. This concept is crucial in various fields, including optimization and control theory. These inequalities can be used to compare matrices, such as determining if one matrix is less than or equal to another in the sense of positive semi-definiteness. This is often denoted as A ≤ B, meaning that B - A is a positive semi-definite matrix. Understanding matrix inequalities is essential for solving problems involving systems of equations and stability analysis.