What are the real life applications of quadratic forms? I have used them to sketch conics but are there any other applications?
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There is a huge number of applications in probability and statistics:
- Modeling "linear" dependence between random variable through second moments (covariances and correlations).
- Linear regression.
- Principal components analysis.
- Assessing precision of estimators.
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Learner
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UPDATED: Looking for a simple intuition: would it be misleading to suggest that quadratic forms are largely (though not exclusively) about distances of one sort or another? Not just Pythagoras, or the metrics of Special and General Relativity, but also in e.g. the distance of linear regression data-points from the optimal line, as mentioned in the answer above? (I’ve seen https://math.stackexchange.com/questions/1899172/intuition-meaning-behind-quadratic-forms?rq=1 and recognise that quadric surfaces etc. are what you get by just setting value of a quadratic form to a constant.) – iSeeker Jul 27 '19 at 14:16
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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.
Mårten W
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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.
Elmar Zander
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