# Problems where SPD linear system arises

I know some of the places where SPD linar systems arises such as elliptic PDEs and normal equations. Can I have a more comprehensive list of scientific applications which require solving SPD linear system. I am specially interested in cases where matrix is not sparse and cases where matrix might not be available directly

Thanks

• I realise you're interested in dense systems in particular, but I though it was worth mentioning that sparse SPD systems often arise in finite element solutions of physical systems because of Newton's 2nd law. If element A acts on element B, then element B acts on element A in exactly the same way, and hence the system is automatically symmetric. There are invariably good physical reasons why the system should be PD, too. – Pseudonym Apr 26 '13 at 5:31
• Similar things hold for matrices arising from circuit theory ( exchange of charges, I give you +1 charge, means you give me -1 charge). However, they are mainly interested in evolution of dynamic system. I am more interested in cases where you solve Ax=b, A being spd (may be as a part of larger problem) – piyush_sao Apr 26 '13 at 6:14

Statistics. Correlation matrices are positive semidefinite. Specifically, let $x$ be a random variable with instances in $\mathbb{R}^n$. Then $E(xx^T)=\Sigma$, where $\Sigma$ is positive semidefinite. Note that it does not matter what distribution the random variable is drawn from; if the covariance matrix exists, it is positive semidefinite. Of course, there are statistical applications in a wide variety of fields.
Control theory. The Lyapunov stability theory says that a sufficient condition for the stability of a system described by a vector ODE $\dot{x}=\phi(x)$ is stable if there exists a positive definite matrix $P$ such that $\tfrac{d}{dt} (x^T P x) < 0$ for all $x$.