Karlins' Inequalities

Karlins' inequalities provide bounds on the difference between a matrix's quadratic form and a scalar multiple of its norm, revealing how matrix entries influence stability and performance in systems. Essentially, they help estimate how far a matrix's behavior can vary, ensuring that certain properties remain within predictable limits. These inequalities are useful in analyzing how small changes in a system or data can affect outcomes, especially in control theory, numerical analysis, and optimization, by quantifying the relationships between matrix elements, eigenvalues, and norms.