# Tagged Questions

Questions about quadratic forms in multiple variables, for example $4x_1^2 + 3x_1x_2 + 5x_1x_3 - 8x_3^2$.

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### Help solving $ax^2+by^2+cz^2+dxy+exz+fzy=0$ where $(x_0,y_0,z_0)$ is a known integral solution

Help solving over the integers: $$ax^2+by^2+cz^2+dxy+exz+fzy=0$$ where $(x_0,y_0,z_0)$ is a known integral solution and $a,b,c,d,e,f$ are integral coefficients. I found in Tito Piezas' identities the ...
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### Linear reformulation or approximation of a quadratic inequality set

Based on the useful comments I reformulated my problem - hopefully it's more clear now. Let $A,B \in \mathbb{R}^{d \times d}$ be symmetric positive-semidefinite matrices, $x \in \mathbb{R}^d$ and ...
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### Lagrange Method of Quadratic Form the a Billinear Form

In the following question I have to present the bilinear form as sum of squares with Lagrange method. $$q(x_1,x_2,x_3,x_4)=2x_1x_4-6x_2x_3$$ However I don't know how I can do it here since none of ...
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### Expectation of Univariate Quadratic Form under Multivariate Gaussian

Is there an obvious trick I am missing for solving the following integral: $$\int_x P(y|x) W(x) (-x^TMx+2x^Tm -c)dx$$ Distributions are Gaussians and $M$ is symmetric. I know how to do the ...
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### How many different values can $(x^2 + y^2, x^2 + 2y^2 )$ have mod 4?

It is known that if a prime number $p = x^2 + y^2$ is equivalent to $p \equiv 1 \mod 4$. This is Fermat's theorem on the sum of two squares. My question is about the value of two simultaneous ...
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### Are all “forms” linear maps from vector spaces to fields?

It seems that whenever we call something a "form": quadratic form, linear form, bilinear form, one-form, two-form, etc. it is always a linear (or perhaps not?) map from some vector space (or ...
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### Why is the inner product of a quadratic form a quadratic form?

I was going through a derivation of the second derivative of the $\log \det X$ where $X$ is symmetric positive definite, I noticed that despite the second order approximation of log det is written as: ...
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### Assume this equation has distinct roots. Prove $k = -1/2$ without using Vieta's formulas.

Given $(1-2k)x^2 - (3k+4)x + 2 = 0$ for some $k \in \mathbb{R}\setminus\{1/2\}$, suppose $x_1$ and $x_2$ are distinct roots of the equation such that $x_1 x_2 = 1$. Without using Vieta's formulas, ...
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### Primes of special form

Are there infinitely many primes $p$ of form $$2^k+a^2=p^2<2^{k+2}$$ where $a\in\Bbb N$? Which primes are known to be of such form? An example is $16+3^2=5^2$. This is the only one I could find.
We know that every prime $1\bmod 4$ can be written in an unique way as $a^2+b^2$ form where $a,b\in\Bbb N$. Is there a comprehensive list of other statements of form "every prime $d\mod r$ can be ...
For each of the following quadratic forms, determine whether the form has a non-trivial zero (we do not need to exhibit it): $f(x, y, z) = 2x^2 + 3y^2 - 6z^2$; \$g(x, y, z) = 2x^2 + 3y^2 - ...