Tagged Questions

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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A specific spanning property of a family of vectors

Let $v_1, v_2, v_3, \dots$ be a family of vectors in $\mathbb R^n$ that span $\mathbb R^n$. Given this family, define the family of vectors \begin{align*} \begin{pmatrix} v_1 \\ \alpha_1 v_1 \end{...
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integral basis of orthogonal complement

Suppose there are $r$ linearly independent vectors $v_1,\dots,v_r\in \mathbb{R}^n$, all of them have integer-valued entries and $\|v_i\|_\infty\leq m$ for some integer $m$. My goal is to find an ...
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Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here. I'm ...
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Independence in mathematics

While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over ...
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A question on linear groups

Asking this question I have made a mistake joining my main question with two simple ones, so it hasn't received enough attention, however there was a partial answer, which was not elaborated, and now ...
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Minimize matrix distance to tensor product

Minimize the following function: $f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are fixed and we minimize over all $V \in SU(n)$. The norm is from the trace inner product. ...
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Determinant of a tensor product [closed]

Let V and W be two vector spaces over a field of characteristic zero. Give a formula for the top exterior power of V tensor W.
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Does a vector belongs to a simplicial subcone when it belong to cone with more than n generators?

Assume $x_{0}\in \text{cone}(a_{1},\dots,a_{N})$, where $a_{i}\in \mathbb{R}^{n}_{+}$ ($a_{i}\in \mathbb{R}^{n}$, and $a_{i}\geq 0$) for $i=1,\dots,N$ (i.e., $x_0$ lies in the cone generated by $a_{i}$...
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Approximate largest eigenvalue of Monodromy matrix

Does anyone know the procedure (or have pseudo code) to approximating the largest eigenvalue of a monodromy matrix? Or even to approximate the monodromy matrix itself? There is no explicit solution ...
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Maximize inner product of a tensor of unitary matrices

How can one maximize the following function: $f(V) = || V \otimes V - U_1 \otimes U_2 ||$ where $U_1, U_2 \in SU(n)$ are given and we seek to maximize over $V \in SU(n)$. Both the maximum value of ...
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How often does a pair of linear maps generate a Zariski-dense subgroup of $GL(d,\mathbb{R})$?

I am an analyst working on a number of problems which in some way relate to random matrix products. In this context I frequently find that the analytic properties I am interested in depend in some way ...
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Existence of Nonnegative Solutions of Linear Systems of Equations and Inequalities with particular constraints

Suppose we have an $n \times m$ nonnegative matrix $A$, where each row sums to $1$. I wonder whether there exists an $m \times n$ nonnegative matrix $X$ that satisfies the following constraints: ...
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Rank of a fat random matrix

Let $\mathbf{R} \in \mathbb{C}^{~n \times k}$ with $n \leq k$ be a random matrix, whose entries are i.i.d zero mean random variables with circularly symmetric Normal distribution. Two questions: (...
Let's suppose we have two variables $(\alpha, \beta)$ on two functions $k_1, k_2$ that can be defined in terms of matrix relations: $$k_1 = \left| \xi^T \mathcal{F}^T \mathcal{F} \xi \right|$$  ...