# 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

## 2 Answers

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 $x$.

Semidefinite programming. There are a variety of useful mathematical models that involve finding a positive semidefinite matrix that satisfies certain linear relationships on its elements, and possibly optimizing a linear function of said matrix. These models are called \emph{semidefinite programs}. See this paper for a discussion of the mathematics involved, and this paper for some applications, including circuit design, structural optimizaton, and others. Search for the word "semidefinite" on this page for even applications. (Disclosure: this was my research group in college.)

Typical (time-)discretizations of parabolic PDEs with a symmetric (self-adjoint) spatial operator also lead to SPD matrices. An example would be the heat equation.

As do (time-)discretizations of hyperbolic equations with self-adjoint spatial operators such as the wave equation.

For full matrices, look at integral equations (or the boundary integral re-formulation of partial differential equations) which typically also lead to symmetric and often positive definite matrices. They are a special case of Fredholm integral equations.