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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### How to Compute the Second Derivative of a Quadratic Maximization Problem with Respect to the Weight Parameter?

I am working on a problem involving the maximization of a function subject to a normalization constraint, specifically in the context of quadratic forms and eigenvalues. The objective function is ...
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### Lower bound for the formula

Is there any formula to find the lower bound or the approximation of the below? $$\left(\frac{y+\frac{k}{y}}{x+\frac{k}{x}} \right)^2 - \left(\frac{y}{x}\right)^2$$
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### Quadratic form with positive semi-definite matrix

Suppose $A$ is a positive semi-definite matrix, and $x$ and $y$ are real vectors. Under what conditions does the following result hold? $$x'y > (<) 0 \Longrightarrow x'Ay \geq (\leq) 0$$ ...
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### Ratio of cubic and quadratic form is approximately normal?

Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which is a ratio of a cubic form and a ...
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### Eigenvalues of a symmetric matrix with known column/row $l^2$-norm

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Suppose we use the notation $a_{i}$ for the $i$-th row (or column, which would be the same given its symmetry), with $i\in\{1, ... , n\}$. ...
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### Let $n\equiv 1\pmod 8$. Do there exist $x,y,z\in\mathbb{Z}$ with $x\equiv \pm3\pmod 8$ and $x^2+4y^2+4z^2=n$?

Let me preface this by saying I know very little about quadratic forms and most of what I know is about quadratic forms in two variables, whilst this question is about a quadratic form in three ...
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### How can I tell if these matrices are congruent?

I am completey lost on this. I know a matrix $B$ is congruent to $A$ if $B = P^\top\!\!AP$ but I tried finding the e-vectors and e-values for $P$ and $P^\top\!$ to spit out $A$ again. I really don't ...
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### Why is the centered moment for this formula so different?

Define the following two functions $Q$ and $S^2$: One of the exercises now asks me to calculate the variance of $S^2$. I tried doing so with the help of a formula that makes use of the third and ...
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### Quadratic Form and its Matrix, Associated bilinear form for positive definite quadratic form is nondegenerate.

I have a question regarding the quadratic forms and their associated matrices. For some reason, google keeps telling me that this matrix is symmetric. The definition I am using is that a quadratic ...
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There is an $\mathbb{R}$-bilinear operation on the octonions $\mathbb{O} \otimes_{\mathbb{R}} \mathbb{O} \rightarrow \mathbb{O}$, which is not associative. My question is instead about the algebra ...
I am interested in the diffeomorphisms $T : \mathbb{R}^n \to \mathbb{R}^n$ that preserve the Euclidian norm, i.e such that $T(x)^\top T(x) = x^\top x$ for all $x \in \mathbb{R}^n$. Do we know how to ...