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schn
schn
12:10 AM
Any insights are appreciated: math.stackexchange.com/questions/3406440/…
 
 
2 hours later…
Martin Sleziak
Martin Sleziak
2:34 AM
0
Q: Diagonal representation of quadratic form to evaluate min. and max. values

schnConsider the problem of finding the minimum and maximum of the quadratic form $$r(\textbf{x})=x_1^2+x_2^2+x_3^2$$ subject to the constraint $q(\textbf{x})=x_1^2+3x_2^2+x_3^2+2x_1x_2-2x_1x_3-2x_2x_3=1$. By finding the eigenvalues of the matrix of $q$, the diagonal representation is $q$ is $y_2^2...

quadratic-forms
(I am just reposting it in a way that's somewhat better readable here in chat - so that other people can have at least some idea about the question without having to click on the link.)
 
Martin Sleziak
Martin Sleziak
2:55 AM
@schn I am not sure whether this is what you're looking for, but $r(\textbf{x})=x_1^2+x_2^2+x_3^2$ remains the same, if the basis is changed based on orthogonal matrix. (I.e., if we do a rotation.)
A real symmetric matrix is diagonalizable using an orthogonal matrix, this is Principal axis theorem.
 

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