# 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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### Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form?

Task: Is the restriction of a Minkowski-form in $\Bbb R^n$ on a vector subspace $U$ with $\dim(U) = n - 1$ also a Minkowski-form? Solution: Since a Minkowski-form has the type $(n - 1, 1)$, ...
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### What are some examples of applications of integral quadratic forms in $n$ variables in algebraic topology?

I'm reading the wiki page of qudratic forms. It simply seems curious to me what are some concrete examples of applications of integral quadratic forms in algebraic topology. I've searched a bit but a ...
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### Confusion of a formula about Lagrangian

Recently, I am reading a paper about eigenvalue problems. Consider the following problem, which occurs at the first page of the paper. \begin{align} \text{minimize}\quad &x^TAx \\ \text{subject ...
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### What do mathematicians mean when they say “form”?

As in differential form, modular form, quadratic form? I'm sorry if this is a really silly question.
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### Reducing a rational ternary quadratic with zeros in its diagonal into a canonical form.

Let the following be a ternary quadratic form: $$A = \begin{pmatrix}0 & a & b \\ a & 0 & c \\ b & c & 0 \end{pmatrix}$$ with $a,b,c\in\mathbb{Q}$. If at least one term in the ...
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### Can the 290 Theorem be refined/sharpened to include special conditions?

The 290 theorem states If a positive-definite quadratic form with integer coefficients represents the twenty-nine integers $1$, $2$, $3$, $5$, $6$, $7$, $10$, $13$, $14$, $15$, $17$, $19$, $21$, ...
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### Is this number positive?

Let $(a_{ij})$ be a collection of non-negative numbers indexed by integers $1\le i,j \le N$ where $N$ is some fixed integer. Let $(c_{ij})$ be another collection of real numbers also indexed by ...
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### If $f,g$ are quadratic forms over $\mathbb{R}$ and $f$ is positive definite, can you reduce the both simultaneously to sum of squares?

If $f,g$ are quadratic forms over $\mathbb{R}$ and $f$ is positive definite, can you reduce the both simultaneously to sum of squares? This question appeared from a friend of mine and I did not ...
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### How two find the matrix of a quadratic form?

I was wondering. If I have a bilinear symmetric form, it is easy to find its matrix. But, when I have a quadratic form, which is the procedure to do that? I heard that one possibility is: If $q$ is ...
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I am struggling with a problem in Shafarevich's Basic Algebraic Geometry. First, some context: Fix $k$ an algebraically closed field. Lines in $\mathbb{P}^3$ correspond to planes through the origin in ...
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### can $4^{2n }$ be written as the sum of “three squares”?

Lagrange theorem says only numbers $n \neq 4^n ( 8k+7)$ can be written as the sum of three squares. what about this one? $$4= 2^2 + 0^2+ 0^2$$ this looks acceptable to me, and yet it is ...
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### show the quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$ is quasi-concave

I have a quadratic function $W(x_1,x_2,\ldots,x_n)=A\sum_{i} x_i^2+ \sum_{i\neq j} x_ix_j$, with $x_i$ nonnegative and $A \in[0,1)$. And w.l.o.g. we can normalize $x_i's$ to between 0 and 1. In ...
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### a Matrix that returns the average of a vector?

Let's concentrate in the 2 dimensional case. I'm looking for a $2 \times 2$ Matrix $A$ that for a vector $x=(x_1, x_2)$ will satisfy: $$xAx^t = \frac{x_1+x_2}{2}$$ Using simple methods of ...
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### System of two quadratic equations in two variables with two parameters leads to quintic polynomial

Actually, it's two closely related systems. Let $a,b \in \mathbb{Q}$ be the parameters. The first system has the form: $$(1+a y)x^2-2(a+y)x+(1+a y)=0 \\ (1-b x)y^2-2(b-x)y+(1-b x)=0$$ One of the ...
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### Finding the parameters of an ellipsoid given its quadratic form

Suppose we have the quadratic form of an ellipsoid of the form $$ax^2 + by^2+cz^2+dxy+eyz+fxz+gx+hy+iz+j=0$$ I want to find centroid of the arbitrarily oriented ellipsoid, its semi-axes, and the ...
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### Constrained optimization problem using Largange multipliers: ellipsoid collision detection and response

This one is purely for the mathematics so the result is far less important than the method itself. My task is to implement a fast and efficient ellipsoid collision detection and response algorithm. ...
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### Partial Derivative of a quadratic form

I want to derive, w.r.t $x$, this: $x'Ax+2y'B'x+y'Cy$ The reference says: "Assuming $A$ positive definite, then the partial derivative is: $2(Ax+By)$." Why the transpose $x'$ it's not in the ...
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### condition for a binary quadratic form to be positive at infinity

I have a sort of binary quadratic form $Q(x,y)=\lambda_1x^4 + \lambda_2 x^2y^2 +\lambda_3 y^4$. I can assume $\lambda_1, \lambda_3 > 0$. Consider $$\int dxdy \; e^{-Q(x,y)}$$ What minimal ...
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### Quadratic form with one value changed.

I came across the following problem when trying to run a Metropolis algorithm. (It is related to computing a multivariate normal density.) Let us have an $n\times n$ matrix $A$ of a special kind: ...
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### Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
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### Prove or disprove: quadratic form $Q(v):=\beta(v,v)$ (biliniarform) is nondegenerate if $\beta$ is.

Let char$K \not = 2$ and $\beta$ be a bilinearform (not necessarily symmetric) on a $K$-vectorspace $V$. Let $Q$ be defined by ($v, w \in V$ arbitrary)$$Q(v) = \beta (v,v)$$ I've shown that $Q$ is a ...
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### Is there a Fermat-era proof of Theorem 69 from Dickson's Intro to NT?

In Dickson’s Introduction to the Theory of Numbers (Ch. VI, pp. 91-93), he gives the following [wonderful and wonderfully general] theorem. Theorem 69: All integral solutions of $$x^2-my^2=zw$$ ...
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### (Geometric) Intuition behind Different Types of Rank 2 Tensor (Specifically Quadratic Forms)

This is essentially a follow-up to this question: Differences between a matrix and a tensor I think I have a good intuition/idea for the change of basis for a rank-(1,1) tensor ($A\vec{v} = \vec{w}$) ...
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### A specific case of quadratic forms

I have a quadric as follows: $$ax^2+by^2+bz^2+yz=0.$$ I am curious to know which shapes in $\mathbb{R}^3$ this equation describes for different value of $a$ and $b$?
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### How to deduce the formula for quadratic form?

I almost every book about quadratic form we can see it described as following function: $$f(x) = \frac{1}{2}x^T A x - b^Tx + c$$ My question is: How can we deduce this formula? I understand, ...
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### show quadratic forms $x^2 + y^2 + z^2$ and $x^2 - y^2 - z^2$ are equivalent over finite fields $\mathbb{F}_p$

Can I show the diagonal matrix (1,1,1) and (1,-1,-1) are equivalent over the finite field $\mathbb{F}_3$ Can I show the quadratic forms $x^2 + y^2 + z^2$ and $x^2 - y^2 - z^2$ are equivalent over the ...
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### How do I convert a quadratic form to a diagonal form?

I don't understand how I should choose the transformations to convert a quadratic form to a diagonal form. Ex: $x_1\cdot x_2 + x_1\cdot x_3 + x_2\cdot x_3$
Let $A$ be an invertible real $n\times n$-matrix, and $q$ be a nondegenerate quadratic form on $\mathbb{R}^n$. Do we have the QR decomposition for $q$ ? In other words : is it true that there exists ...