What is the importance of definite and semidefinite matrices?

Question

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

Answer 1

There are many uses for definite and semi-definite matrices. I can give just a few examples although undoubtedly I will be missing many.

1. Positive-definite matrices are the matrix analogues to positive numbers. It is generally not possible to define a consistent notion of "positive" for matrices other than symmetric matrices. As a consequence, positive definite matrices are a special class of symmetric matrices (which themselves are another very important, special class of matrices). It turns out that many useful matrices fall under this class such the covariance matrix, overlap matrices used in quantum chemistry and dynamical matrices used in calculation of molecular vibrations (which is positive semi-definite).

2. Definiteness is a useful measure for optimization. Quadratic forms on positive definite matrices
   $$\mathbf{x}^\mathrm{T}A\mathbf{x}$$ 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.

3. Perhaps equally (or more) important, especially to a mathematician, is the fact that the theory of (semi)definite matrices is an incredibly rich and beautiful field. There are chains of elegant results concerning these matrices, especially for positive-definite matrices. That is motivation enough.