# 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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### Quadratic form with one value changed.

I came across the following problem when trying to run a Metropolis algorithm. (It is related to computing a multivariate normal density.) Let us have an $n\times n$ matrix $A$ of a special kind: ...
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### Positive definite quadratic form defines nondegenerate positive inner product

The background to this question (which is not important for the actual question!) is that I'm working on something in geometric dynamics, specifically the Jacobi metric. To me it seems that one uses ...
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### 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 ...
### 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 $\vec x$...