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I can think of (at least) two ways of interpreting this question. First: You are given some specific list of $k$ points $x_1$, $x_2$, ..., $x_k$ in $\mathbb{R}^n$, and you want to detect whether $k$ …

This isn't true if $n$ is odd. For example, if $n=3$, then your formula is $(a,b,c) = (\cos^2 \theta, 2 \sin \theta \cos \theta, \sin^2 \theta)$ and it always lies in the hyperplane $a+c=1$. More gene …

If you have a lot of duplicate rows (and you know what they are), you can reduce to a smaller matrix. I'll start with an example, because writing out the general case will be notationally annoying. L …

What you are seeing is that the orthogonal matrices of determinant $-1$ swap the two spin representations. The first several parts of this argument will be valid for $(4n+2) \times (4n+2)$ matrices as …

A bivector is an element of $\bigwedge^2 V$, so it is dual to a $2$-form on $V$. You can think of a bi-vector as a tiny piece of area. If $V$ is three dimensional and comes with an inner product, th …

Another proof: For any $I$ and $J$ two subsets of $\{1,2,\ldots,n\}$ of the same cardinality, let $D(I,J)$ be the minor in rows $I$ and columns $J$. Let $Pf(I)$ be the Pfaffian $\sqrt{D(I,I)}$. We set …

Here is a simple idea, with no complexity analysis. Compute a basis $v_1$, ..., $v_{n-r}$ for the kernel of $A$; this can be done with exact arithmetic in $n^3$ operations by Gaussian elimination. Com …

Let $J_4(n)$ be the $n \times n$ matrix $$\begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 & \cdots \\ -1 & 0 & 0 & 0 & 0 & 0 & \cdots \\ 0 & 0 & 0 & 1 & 0 & 0 & \cdots \\ 0 & 0 & -1 & 0 & 0 & 0 & \cdots \\ 0 …

Partial progress: It's easy to achieve $n-3$. Consider matrices of the form $$\begin{pmatrix} 0 & 0 & r_1 & r_2 & \cdots & r_{n-3} & 0 \\ 0 & 0 & 0 & r_1 & \cdots & r_{n-4} & r_{n-3} \\ r_1 & 0 & & …

Your first condition implies that $r$ is the order of $p$ modulo $q$. If the order of $1-t$ is $q$, then $p$ divides $$ \prod_{i=1}^{p-1} \prod_{j=1}^{p-1} (1 - \zeta^i - \zeta^j) \quad (*)$$ where …

In $\mathbb{R}^3$, Milenkovic and Milenkovic give an alogrithm for efficiently approximating an orthogonal matrix by a rational orthogonal matrix. As lhf points out, the inverse of an orthogonal matri …

OK, here might be an answer to the question you are meaning to ask: Let $a_1$, ..., $a_n$ be unit vectors in $\mathbb{R}^d$. Let $G$ be a group acting linearly on $\mathbb{R}^d$, which permutes the $ …

I haven't read this paper, but it looks relevant: Principal nilpotent pairs in a semisimple Lie algebra I. To quote from the MathSciNet review: The author notes that the general problem of clas …

The inverse of a matrix is the adjoint divided by the determinant. So what you want to compute is the determinant of an $(n-1) \times (n-1)$ submatrix, divided by the determinant of your original matr …

If it is clear from context that you are working in the ambient setting of $k[x,y]$, then you can write $k[x] + k[y]$. Otherwise, I would spell it out in words.