TITLE: What is the importance of definite and semidefinite matrices?
QUESTION [14 upvotes]: 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

REPLY [4 votes]: Positive definite matrices have applications in various domains like physics, chemistry etc. In CS, optimization problems are often treated as quadratic equations of the form $Ax=b$ where $x$ can be any higher degree polynomial. To solve such equations, we need to calculate $A^{-1}$ which is then used to find out $x=A^{-1}.b$.
Computing $A^{-1}$ is time consuming for complex higher rank matrices. Instead we use Cholesky’s decomposition: $A = L.L^T$, if the matrix $A$ is symmetric, positive definite then we can decompose it into lower triangular matrix $L$. Finding the inverse of this lower triangular matrix $L$ and its transpose $L^T$ is computationally efficient process. Hence we get huge performance gain on $x = (L^T)^{-1}.(L^{-1}).b$