# 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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### 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 ...
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### 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 ...
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### 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 $Y^2 - X^2 - X Z = k Z^{(2)}$ ...
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### 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 ...
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### Integral solutions of hyperboloid $x^2+y^2-z^2=1$

Are there integral solutions to the equation $x^2+y^2-z^2=1$?
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### Should isometries be linear?

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-...
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Let $\Gamma = SL_2(\mathbb{Z})$. Let $\mathcal{H} = \{z \in \mathbb{C}\ |\ \mathcal{Im}(z) > 0\}$ be the upper half complex plane. Let $\sigma = \left( \begin{array}{ccc} p & q \\ r & s \... 3answers 3k views ### Hessian matrix of a quadratic form I need a help with one example. I have to proove that hessian matrix of a quadratic form$f(x)=x^TAx$is$f^{\prime\prime}(x) = A + A^T$. I am not even sure how the Jaxobian looks like (I never did ... 3answers 502 views ### Etymology of the word “isotropic” Given a quadratic form$q : V \rightarrow k$, a nonzero vector$v \in V$is said to be isotropic if$q(v) = 0$. Any subspace of$V$containing such a vector is also said to be isotropic, and the ... 1answer 260 views ### A condition for an odd prime to be represented by a binary quadratic form of a given discriminant Let$f = ax^2 + bxy + cy^2$be an integral binary quadratic form. We say$D = b^2 - 4ac$is the discriminant of$f$. If$D < 0$and$a > 0$, we say$f$is positive definite. It is easy to see ... 2answers 2k views ### non-symmetric positive definite matrix!? Is symmetry a necessary condition for positive (or negative) definiteness? If not: It can be proved that if$\mathbf{A}:(m\times m)is a square (non-symmetric) matrix, then \mathbf{z'Az=z'Bz},~~... 1answer 214 views ### finding if two binary quadratic forms represent the same integers I am currently working with quadratic forms for a given discriminant D; to get all primitive forms (one for each equivalence class) I found this website : http://www.numbertheory.org/php/classnoneg.... 2answers 1k views ### Meaning of the identity \det(A+B)+\text{tr}(AB) = \det(A)+\det(B) + \text{tr}(A)\text{tr}(B) (in dimension 2) Throughout, A and B denote n \times n matrices over \mathbb{C}. Everyone knows that the determinant is multiplicative, and the trace is additive (actually linear). \begin{align*} \det(AB) = \... 2answers 773 views ### Generalizing the 290 theorem. I have only just come across the remarkable theorem of Conway about universal quadratic forms over \mathbb{Z}; namely that in determining whether a integer coefficient, positive definite quadratic ... 0answers 1k views ### Simultaneous diagonalization of quadratic forms I would like to collect references (or direct quotations) about as many "simultaneous diagonalization" results in linear algebra as possible. Let V be an n-dimentional (n finite) vector space ... 1answer 120 views ### On a remarkable system of fourth powers using x^4+y^4+(x+y)^4=2z^4 The problem is to find four integers a,b,c,d such that,a^4+b^4+(a+b)^4=2{x_1}^4\\a^4+c^4+(a+c)^4=2{x_2}^4\\a^4+d^4+(a+d)^4=2{\color{blue}{x_3}}^4\\b^4+c^4+(b+c)^4=2{x_4}^4\\b^4+d^4+(b+d)^4=2{x_5}... 1answer 195 views ### On products of ternary quadratic forms\prod_{i=1}^3 (ax_i^2+by_i^2+cz_i^2) = ax_0^2+by_0^2+cz_0^2$The equation, $$(ax_1^2+by_1^2)(ax_2^2+by_2^2) = ax_0^2+by_0^2\tag1$$ has the well-known solution when$a=b=1$, $$(x_1^2+y_1^2)(x_2^2+y_2^2) = (x_1 y_2 + x_2 y_1)^2 + (x_1 x_2 - y_1 y_2)^2$$ ... 2answers 9k views ### Find the EigenValues and EigenVectors of the matrix associated with quadratic form Problem: Find the EigenValues and EigenVectors of the matrix associated with quadratic forms$2x^2+6y^2+2z^2+8xz$. I know how to convert a set of polynomial equations to a matrix but I have no clue ... 1answer 97 views ### What are numbers$n$such that$a^2+nb^2 = c^2$and$na^2+b^2 = d^2$? Let$n$and$a,b,c,d,$be in the positive integers. I. For the system, $$a^2-nb^2 = c^2\\a^2+nb^2=d^2$$ then$n$is a congruent number. The sequence starts as$n=5,6,7,13,14,15,20,21,$and so ... 1answer 75 views ### Pell's equation and representation elements of$\mathbb Z_p$. We defined the function$f:\mathbb Z_p \times \mathbb Z_p \rightarrow \mathbb Z_p$such that$f(x,y)=x^2-cy^2$and$c\not\equiv 0\pmod{p}$. Is it true that$f$is onto? 4answers 153 views ### 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? 2answers 97 views ### On$p^2 + nq^2 = z^2,\;p^2 - nq^2 = t^2$and the “congruent number problem” (Much revised for brevity.) An integer$n$is a congruent number if there are rationals$a,b,c$such that, $$a^2+b^2 = c^2\\ \tfrac{1}{2}ab = n$$ or, alternatively, the elliptic curve,$$x^3-n^2x = ... 4answers 646 views ### How can I find integer values for which a given expression gives a perfect square? Find the integer values for which$x^2+19x+92$is a perfect square. Also, How to proceed if you have to find values ( not necessarily integer)? 1answer 150 views ### A problem on positive semi-definite quadratic forms/matrices Suppose$a+b+c=0$and (without loss of generality)$a\leq b\leq 0\leq c$,$a^2+b^2+c^2=1$, is the following quadratic form positive semi-definite? Thank you very much. \begin{equation*} \begin{split} ... 4answers 5k views ### Determining if a quadratic is always positive Is there a quick and systematic method to find out if a quadratic equation is always positive or may have positive and negative or always negative for all values to its variables? Say for a quadratic ... 1answer 200 views ### Expressing a quadratic form,$\mathbf{x}^TA\mathbf{x}$in terms of$\lVert\mathbf{x}\rVert^2$,$A$EDIT: This question is actually an attempt to solve this. Please take a look. Let$A$be a symmetric postive-definite$n\times n$matrix, i.e.$A\in\mathbb{S}_{++}^{n}$Also, let$\mathbf{x}\in\...
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I would like to solve the following optimisation problem: $\min_{x} (x'Ax)$ subject to $x'Bx = x'Cx = 1$. Where A is symmetric and B and C are diagonal. Does anyone have a suggestion for an ...
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### Proving the multiplicativity of a binary quadratic form

Consider the set $S$ of all integers of the form $x^2+y^2+4xy$, where $x$ and $y$ are integers. How could one prove the set $S$ is closed under multiplication? I have tried the bashy brute force ...