# Tagged Questions

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### Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial

The following question is motivated by the study of a stability border for a robust linear time-invariant control system. Let us we have an affine family of $n\times n$ matrices with indeterminate ...
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### Powers of linear functions span the space of polynomial functions?

Let $P_n$ be the space of polynomials in $n$ variables over a field of characteristic 0. I'm very sure that the space spanned by powers of linear functions is the whole space $P_n$. Anyone can come ...
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### Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial $$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n}$$ After the linear change of ...
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### Union of subspaces not contained in algebraic variety

Consider an $n \times n$-matrix $A$ and an $m \times n$-matrix $C$ where $m < n$. I am asking myself whether it is possible to formulate an algebraic condition on $A$ and $C$ such that the union ...
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### Paralel bezier curve

If I have a cubic Bezier curve specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined the ...
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### Opposite complex structure on Kaehler manifold

Hallo, Let $(M,J)$ be a Kaehler manifold. How can one descride the opposite complex structure? What is the precise definition of the opposite complex structure? Can one descride the opposite complex ...
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### How to determine there exists a unique invariant subspace for a set of matrices

Hi everyone, Ive been looking at the following problem, but its not entirely in my area and some potential solutions seem to rely on algebraic geometry. Maybe thats just a complicated way to solve ...
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### Multivariate analogue of Vandermonde determinant

Dear all, Consider the $(n+1)\times (n+1)$ matrix $A$ with indeterminates $X_i, Y_i$, $0\leq i\leq n$ such that the $(i,j)$-th entry is given by $X_i^jY_i^{n-j}$. The $i$-th row is ...
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### Optimization over Spectral Laplacian in cycles and trees

Is there any idea on how one can deal with an optimization problem of sum of k largest eigenvalues(min) of Laplacian matrix of a simple cycle or tree? I would like to use semidefinite programming for ...
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### Arrangements of graphs of linear functions

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. Assume $X$ is unbounded. A function $f:X\longrightarrow\mathbb{R}$ is linear if it is of the form $$f(\bar{x})=a_1x_1+\ldots+a_nx_n+b$$ for some ...
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### Arrangements of hyperplanes

Fix $n>0$ and $X\subseteq\mathbb{R}^n$. A function $f:X\longrightarrow\mathbb{R}$ is linear if it is of the form $$f(\bar{x})=a_1x_1+\ldots+a_nx_n+b$$ for some $a_i,b\in\mathbb{R}$. Suppose we ...
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### General Orthogonal Group and its properties

I know that exist a Lie Group Called the Orthogonal Group $O(n)$. That correspond to all matrix of $n \times n$ in the real numbers such that the columns are a orthogonal basis for $\mathbb{R}^n$. Is ...
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### When is a matrix similar to a non-negative matrix?

Consider a real square matrix $A$ of size $n\times n$. Under which conditions on $A$ does there exist a row-stochastic matrix $U$ (non-negative, rowsums = 1), such that $A'=U^{-1}AU$ is a non-negative ...
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Hi, let $L$ be a lattice isomorphic to $\mathbb{Z}^r$ for some positive integer $r$ and $E=L\otimes \mathbb{R}$. An integral paving in $E$ is a set $\Sigma$ of integral polytopes (the vertices are ...
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### Matrices whose kernel escapes a sub-vector space

Let $n>n'\gg m$ and $V$ be a subspace of $\mathbb{C}^n$ of dimension $n'$. I am trying to characterize the set $X$ of $m\times n$ matrices $A=(a_{ij})$ satisfying $\ker(A)\not\subseteq V$, that is, ...
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### best rank r approximation for non-Frobenius norm

The best rank $r$ approximation to a given matrix $M$ in Frobenius norm, according to Eckart-Young theorem, is truncated SVD - just keep $r$ largest singular values. What if I need to construct best ...
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### Classification of Tori of GL2, up to conjugation

Over an algebraically closed field $k$, every one-dimensional torus embedded (as a closed algebraic subgroup) into GL2 is diagonalisable, and the embedding is $t\mapsto (t^m,t^n)$ for some integers ...
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### Solution Existence of a System of Complex Quadratic Equations

Consider ${x_k}_{k = 1}^K \in \mathbb{C}^{N \times 1}$ a set of $K$ complex vector variables of length $N$. I am interested in finding the existence of a solution of the following quadratic set of ...
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### Factorization of bivariate polynomial

Let $q(y, z) = u_1 + u_2y + u_3 z + u_4y^2 + u_5yz + u_6z^2 + u_7y^3 + u_8y^2z + u_9yz^2 +$ $\hspace{2.55cm}u_{10}y^3z + u_{11}y^2z^2 + u_{12}y^3z^2$ Can $q(y, z)$ be factorized as ...
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### Systems of simultaneous real quadratic equations

Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of ...
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### On the Positive Definiteness of a Linear Combination of Matrices

In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated. QUESTION: Let $m,n\in\mathbb{N}$ and let $A_1,\ldots, A_m\in M_n(\mathbb{R})$ be real, ...
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### Real Pfaffian representations of real cubic surfaces

Consider the following classical construction (which is called Pfaffian representation as Sasha indicates): Let $V^4\subset \Lambda^2 \mathbb R^6$ be a four-dimensional subspace of the space of ...
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### Impossibility of continuously picking k independent rows from a rank k matrix

Suppose I have an $n\times n$ real (or complex) matrix of rank $k$, and I want to pick $k$ linearly independent rows from it. I want to do this in a continuous fashion as the matrix varies ...
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### Subspaces of End(V) that can fix any vector

Suppose V is a finite-dimensional vector space and I have a linear subspace of its endomorphisms $$W \subseteq \mbox{End}(V).$$ How can I easily check if every vector of $V$ is fixed by some element ...
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### Explicit equation of Dickson invariant / quasideterminant / special orthogonal group over the integers

Consider $2n$ coordinates $x_1,\ldots,x_n,y_1,\ldots,y_n$ and the quadratic form $q = \sum_{i=1}^n x_i y_i$. Now call $O(q,A)$ (orthogonal group of $q$) the group of $(2n)\times(2n)$ matrices, with ...
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### Approximating commuting matrices by commuting diagonalizable matrices

Suppose the matrices $A$ and $B$ commute. Do there exists sequences $A_n$ and $B_n$ of matrices such that $A_n \rightarrow A$, $B_n \rightarrow B$. Each $A_n$ is diagonalizable and the same for ...
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### Expectation of a multivariate Gaussian over a plane

For a vector $X$ which follows a multinomial Gaussian distribution $N(\vec{0},\Sigma)$, a given vector $b$, and a known scalar value $c$, I would like to calculate the expectation : $E[X|X^Tb = c]$ ...
I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose sum is zero: $$v_1 + v_2 + \cdots + v_n = 0 \; .$$ Now I form the closed polygon $P$ in space by placing them head to tail. So the ...