# What is the importance of definite and semidefinite matrices?

I would like to know some of the most important definitions and theorems of definite and semidefinite matrices and their importance in linear algebra. Thanks for your help

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$$\mathbf{x}^\mathrm{T}A\mathbf{x}$$ are always positive for non-zero $\mathbf{x}$ and are convex. Analogous results hold for negative-definite matrices. This is a very desirable property for optimization since it guarantees the existences of maxima and minima. It is properties like these for example, that allow you to use the Hessian matrix to optimize multivariate functions.