# 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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### Compute the dimension of the space of quadratic forms

We were asked the following: "Compute the dimension of the space of quadratic forms on $V=\mathbb{R^2}.$ Compute also the dimension of the space of symetric forms on $\mathbb{R^2}$, $S^2\mathbb{R^2}$....
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### How can I find values for which a given expression gives a perfect square?

There have been several posts on this topic on math.se, such as this one with the same title. However all the posts I found contained coefficients to $x^2$, that were perfect squares. I am looking for ...
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### Quartic polynomial in ten variables

I have a quartic form, i.e. a homogeneous 4-th degree polynomial, in ten real variables and the inequality: $f(x_1,\ldots, x_{10}) \geq c$, for some $c>0$, which I believe that geometrically ...
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### $q_I$ primitive as a quadratic form? [closed]

Let $I \subset \mathcal{O}_k$ be an ideal, $N(I) = [\mathcal{O}_K : I] = |\mathcal{O}_K/I|$. Define $q_I$ be $q_I(x) = N_{K/\mathbb{Q}}(x)/N(I)$. Is $q_I$ primitive as a quadratic form?
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### Optimizing the sum of powers of positive quadratic functions

In my research I have come across the following optimization problem. \begin{array}{c} maximize \hspace{1cm} \sum \limits_{n=1}^{N}\left(\mathbf{x}^{T}A_n \mathbf{x}\right)^{k}\\ s....
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### Matrix of quadratic form (in Serre's general notion)?

I am currently reading Serre's book on arithmetic. In chapter four (page 27) he defines a general notion of the quadratic form as: Let $V$ be a module of a commutative ring $A$. A function $Q$ is ...
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### Solution of quadratic diophantine equations

Is there any algorithm so that solution to the following equation can be found? $(x+a)^2-y^2=c$ where $c$ and $a$ is a constant. It is similar to Pells eqution with a variation where $D=1$. I am new ...
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### Quadratic form in Hilbert space associated with orthogonal projection operator

we are in Hilbert space $L^2$ and we are given subspace of dimension $2K$ $$V=Vect\{ g_k,\bar{g_k},1\le k\le K \}$$ $V$ is a sum of $K$ subspaces of dimension 2 $$W_k=Vect \{g_k,\bar{g_k} \}$$ now ...
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$x=(x_1,...,x_T)'$ is a $T\times1$ random vector, where $x_t, t=1,..., T$, is a stationary process with mean zero and finite fourth moments. $A$ is a $T\times T$ symmetric constant matrix. How to find ...
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### Transform $f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$ to a diagonal form.

I try to transform Transform $$f(x_1,x_2,x_3)=2{x_1}^2+5{x_2}^2+5{x_3}^2+4x_1x_2-4x_1x_3-8x_2x_3$$ to a diagonal form. I can do it using eigenvalue, but when I directly complete the square to find its ...
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### Integer solutions to $x^2 + dy^2 = c$

I'm trying to find all integer solutions of an equation $x^2 + dy^2 = c$ with $d,c \in \mathbb{Z}_{>0}$. I'm well aware of the methods that exists when $d \in \mathbb{Z}_{<0}$ or when $c$ is a ...
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### 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 ...
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### How to compute the matrix $S$ in Sylvester's law of inertia

Sylvester's law of inertia states that for any symmetric matrix $A$ there exist an invertible matrix S such that, $S^T A S = D$, where $D$ is a diagonal matrix which has only entries 0, +1 and −1 ...
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### Can I find a vector with entirely nonzero entries for the following quadratic form to evaluate to zero?

I have a square, symmetric matrix $M$, of size at least $2\times 2$, with diagonal entries equal to $1$ and off-diagonal entries equal to $\pm 1$. Let the entries of $M$ be such that it is indefinite. ...
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### Omission in Jacobson's BAI regarding extension of isometries.

Suppose $V$ is a finite dimensional vector space over a field of characteristic $\neq 2$ equipped with a nondegenerate quadratic form $Q$. Witt's cancellation theorem says that if $U_1,U_2$ are ...
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