MHB Quadratic Forms: Beyond Sketching Conics

matqkks
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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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One of the more important (fundamental) optimization problems in control theory is the linear-quadratic regulator problem, in which you try to solve a linear DE subject to a quadratic constraint. The quadratic constraint involves quadratic forms.
 
I have used quadratic forms to match the digital image of a pupil (in perspective) with an ellipse.

A friend of mine worked on an application of quadric forms (3 dimensional) to match a heart in an MRI scan (3D image) with an ellipsoid.
 

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The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
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When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
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