# Tagged Questions

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

240 views

### reference for linear algebra books that teach reverse Hermite method for symmetric matrices

January 13, 2016: book that does this mentioned in a question today, Linear Algebra Done Wrong by Sergei Treil. He calls it non-orthogonal diagonalization of a quadratic form, calls his first method ...
172 views

### Find the transitional matrix that would transform this form to a diagonal form.

Let the quadratic form $F(x,y,z)$ be given as below $F(x,y,z)=2x^2+3y^2+5z^2-xy-xz-yz$ Find the transitional matrix that would transform this form to a diagonal form. I got the symmetric ...
1k views

### How to solve inhomogeneous quadratic forms in integers?

If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ ...
2k views

95 views

### Finding integers of the form $3x^2 + xy - 5y^2$ where $x$ and $y$ are integers, using diagram via arithmetic progression

So the diagram drawn looks like this: We begin at the edges labeled $3$ and $-5$ because we are using those as the bases for $x$ and $y$, respectively. The way we obtain the values of the 2 ...
540 views

### Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
121 views

I'd like to write $2xy+2xz+2yz$ in the form $a(\cdots)^2+b(\cdots)^2+c(\cdots)^2$ where each blank space is a linear combination of $x,y,z$. The closest I have is: $$(x+y+z)^2-(x-z)^2-y^2=2xy+4xz+2yz... 1answer 5k views ### sum of squares of dependent gaussian random variables Ok, so the Chi-Squared distribution with n degrees of freedom is the sum of the squares of n independent gaussian random variables. The trouble is, my gaussian random variables are not independent. ... 5answers 1k views ### Every integer vector in \mathbb R^n with integer length is part of an orthogonal basis of \mathbb R^n Suppose \vec x is a (non-zero) vector with integer coordinates in \mathbb R^n such that \|\vec x\| \in \mathbb Z. Is it true that there is an orthogonal basis of \mathbb R^n containing \vec x... 0answers 238 views ### What numbers are integrally represented by 4 x^2 + 2 x y + 7 y^2 - z^3 EDIT: a selection of material relevant to this problem is available at INHOMOGENEOUS In December 2010 my question appeared in the M.A.A. Monthly, show that 4 x^2 + 2 x y + 7 y^2 - z^3 \neq \pm 2 m^... 3answers 224 views ### Solving a^2+3b^2=c^2 I'm looking for how to solve the equation a^2+3b^2=c^2 where a,b,c are integers and b is even, I'm looking for the algorithm used to solve this kind of equations, not just the solution. Regards 3answers 282 views ### Help solving ax^2+by^2+cz^2+dxy+exz+fzy=0 where (x_0,y_0,z_0) is a known integral solution Help solving over the integers:$$ax^2+by^2+cz^2+dxy+exz+fzy=0$$where (x_0,y_0,z_0) is a known integral solution and a,b,c,d,e,f are integral coefficients. I found in Tito Piezas' identities the ... 1answer 54 views ### Transforming quadratic forms, how is this theorem called? In my textbook there is the following nameless theorem: Let Q=\sum_{i,j=1}^n a_{ij}X_i X_j with a_{ij}=a_{ji}\in K be a quadratic form in n variables over a field K not of characteristic ... 0answers 69 views ### Simultaneously diagonalise two real quadratic forms I would like to simultaneously diagonalise the quadratic forms A=2x^2+3y^2+3z^2-2yz, and B=x^2+3y^2+3z^2+6xy+2yz-6zx. Of course there's a theorem saying this is possible and I followed the ... 1answer 536 views ### How to solve an equation of the form ax^2 - by^2 + cx - dy + e =0? I am trying to find out how to solve ax^2 - by^2 + cx - dy + e = 0 to get integer solutions, failing this the rational solutions. Thanks! 4answers 191 views ### Parametrization of x^2+ay^2=z^k, where \gcd(x,y,z)=1 x,y,z be three coprime integers, a \in \mathbb{Z}>0 and k an odd integer. How do I find all the non-trivial solutions of the diophantine equation?$$x^2+ay^2=z^k$$Does the method which ... 1answer 111 views ### Elementary properties of integral binary quadratic forms Let f = ax^2 + bxy + cy^2 be a binary quadratic form over \mathbb{Z}. D = b^2 - 4ac is called the discriminant of f. We say f is positive definite if a \gt 0 and D \lt 0(cf. this ... 2answers 71 views ### Using Lagrange's diagonalization on degenerate linear forms Let A=\begin{pmatrix}1 & 2 & 3\\ 2 & 3 & 4\\ 3 & 4 & 5 \end{pmatrix} be a real matrix. Find an invertible matrix P\in M_{3}(\mathbb{R}) such that P^TAP is diagonal ... 1answer 2k views ### What's a BETTER way to see the Gauss's composition law for binary quadratic forms? There is a group structure of binary quadratic forms of given discriminant d: Let [f]=[(a,b,c)], [f']=[(a',b',c')], where d=b^2-4 a c=b'^2-4 a' c'. The composition of two binary quadratic ... 2answers 858 views ### Elementary proof that 2x^2+xy+3y^2 represents infinitely many primes? We did in class x^2+y^2, which was easy, and we had for homework 2x^2+2xy+3y^2, which I did (its values (if not square) must be divisible by form primes, or of the form x^2+5y^2, and clearly ... 4answers 621 views ### Representing a number as a sum of at most k squares Fix an integer k >0  and would like to know the maximum number of different ways that a number n can be expressed as a sum of k squares, i.e. the number of integer solutions to$$ n = x_1^2 + ...
Question Suppose $V$ is a (finite-dimensional) vector space over $F$ ($\operatorname{char }F\neq2$, due to user1551) equipped with a non-degenerate quadratic form $Q$, and $T$ is a distance-...