What are the real life applications of quadratic forms? What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
 A: There is a huge number of applications in probability and statistics:


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*Modeling "linear" dependence between random variable through second moments (covariances and correlations).

*Linear regression.

*Principal components analysis.

*Assessing precision of estimators.

*...

A: Just an example: you can understand the conjugate gradient algorithm for solving the linear system $Ax=b$ with $A$ symmetric and positive definite also by way of minimising the bilinear form $\frac12 x^TAx-b^tx$. This has no implication on the algorithm itself, but gives you more insight into what's going on inside (see e.g. http://www.math.nyu.edu/faculty/greengar/painless-conjugate-gradient.pdf). This "dual" view, i.e. solving a linear system vs. minimising a quadratic form, can also be found often in the theory of Galerkin methods  for elliptic PDEs.
A: Among other things, some of which are mentioned in the other answers, you can use them to determine the character of stationary points. If you have a function $f:\mathbb{R}^n\longmapsto\mathbb{R}$, and find points where $\nabla f=0$, then the quadratic form tells you whether it is a saddle point, a min or a max.
