# 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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### Linear transformation of positive definite diagonal matrix

Let $\mathbf \Psi$ denote the set of all positive definite, diagonal, nXn dimensional, real-valued matrices . Let $\mathbf \Phi$ denote the set of all positive semi-definite, diagonal, nXn dimensional,...
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We know convex combination of concave/convex functions are concave/convex. While convex combination of two quasi-convex/quasi-concave functions necessarily quasi-convex/quasi-concave. Common ...
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### Describe an equation geometrically

Finishing the last few stuff left for my end-term semester exams on Linear Algebra II, I bumped across a collection of identical exercises, posting one below : Describe geometrically, giving as much ...
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### How do I convert a quadratic form to a diagonal form?

I don't understand how I should choose the transformations to convert a quadratic form to a diagonal form. Ex: $x_1\cdot x_2 + x_1\cdot x_3 + x_2\cdot x_3$
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### How to deduce the formula for quadratic form?

I almost every book about quadratic form we can see it described as following function: $$f(x) = \frac{1}{2}x^T A x - b^Tx + c$$ My question is: How can we deduce this formula? I understand, ...
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Consider quadratic form $$Q(w,x,y,z)=w^2-x^2-y^2+z^2$$ and fix $r\in(0,\frac12)$ and pick a large enough $n\in\Bbb N$. How do we find a solution to $$Q(w,x,y,z)\bmod n=0$$ on condition that $$\sqrt n\... 0answers 33 views ### QR decomposition for nondegenerate quadratic form Let A be an invertible real n\times n-matrix, and q be a nondegenerate quadratic form on \mathbb{R}^n. Do we have the QR decomposition for q ? In other words : is it true that there exists ... 5answers 1k views ### Every integer vector in \mathbb R^n with integer length is part of an orthogonal basis of \mathbb R^n Suppose \vec x is a (non-zero) vector with integer coordinates in \mathbb R^n such that \|\vec x\| \in \mathbb Z. Is it true that there is an orthogonal basis of \mathbb R^n containing \vec x... 1answer 20 views ### Find an orthogonal Matrix to a quadric Given the following Quadric$$F_4 := \{X \in R^3 | x_1x_2+x_1x_3+x_2x_3 =4\} $$My task is find an orthogonal Matrix C and d_1,d_2,d_3 \in R  so that$$F_4 = C*\{Y \in R^3 | d_1y_1^2 +d_2y_2^2+...
How to check whether the two quadratic forms $$x_1^2 + x_2^2 \quad \text{(I)}$$ and $$2x_1x_2 \quad \text{(II)}$$ are equivalent on each of ...
The Boolean Quadratic Programming problem is defined as: $\min_{x} f(x) = x^TQx + c^Tx$ s.t. $x \in \{0,1\}^n$ It is a well-studied NP-Hard problem with many approximation algorithms proposed. I ...