Quadratic forms are homogeneous quadratic (degree two) polynomials in $n$ variables. In the cases of one, two, and three variables they are called unary, binary, and ternary. For example $\quad Q(x)=2x^2\quad$ is called unary quadratic ploynomial, $\quad Q(x,y)= 2x^2+3xy+2y^2\quad$ is called binary quadratic polynomial and $\quad Q(x,y,z)=2x^2+3y^2+z^2+7xy+5yz+9xz\quad$ is called ternary quadratic polynomial.

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### Matrix quadratic forms with non-symmetric shape parameter

A matrix quadratic form can be expressed as $\mathbf{Q}=\mathbf{XAX}^T$, where $\mathbf{X} \in \mathbb{R}^{n \times m}$ is a matrix and the shape parameter $\mathbf{A}\in \mathbb{R}^{m \times m}$ is a ...
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### Primes representable by either $x^2+36y^2$ or $4x^2+9y^2$ [closed]

Is there a simple criterion for primes that are representable by either $x^2 + 36 y^2$ or $4x^2 + 9y^2$? This is not my area of expertise, so any pointers appreciated. I had a look in the Cox book &...
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Let $F(x_1,\ldots,x_{r+1}) = \sum_{1\leq i,j \leq r+1}A_{ij}x_ix_j$ be an integral quadartic form with rank $r$ such that $A_{11} = 0$ and $A_{12} \neq 0$. Show that there exists a unimodular ...
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### Show that $(x-a)^T(x-a)=\text{tr}(x_c, x_c)+n(a-\bar x)^2$

Show that $(x-a)^T(x-a)=\text{tr}(x_c, x_c)+n(a-\bar x)^2$ (Gentle Matrix Algebra exercise 3.2) This shows that the norm $\|x-a\|$ is minimized when $a=\bar x$. I went through the pages and got the ...
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### Over a finite field, which square matrices produce a zero quadratic form?

For which matrices $A \in (\mathbb{F}_p)^{n \times n}$ do we have $x^T A x=0$ for all $x \in (\mathbb{F}_p)^n$? Obviously, this is the case if $A=B-B^T$ for some $B$ (which is equivalent to saying ...
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