# Tagged Questions

**4**

votes

**1**answer

154 views

### $n$ columns of a specific “infinite” Vandermonde matrix always linearly independent? [on hold]

Consider the "infinite" Vandermonde matrix
$$
V (x_1, x_2, \ldots , x_n) =
\begin{pmatrix}
1 & x_1 & x_1^2 & \cdots & x_1^{n-1} & x_1^n & x_1^{n+1} & \cdots \\
...

**-1**

votes

**0**answers

60 views

### Why solving a system of linear equation produces the intersection of the equation [closed]

Let us consider two equations
1)x+y=1
2)-x+y=1
Consider the solution of the equations using Gaussian Elimination,
Geometrically I am able to understand the ...

**8**

votes

**2**answers

329 views

### Is there a hyperplane avoiding two independent sets?

Let $V$ be a vector space over a field with $5$ elements, $A,B \subseteq V$ independent subsets. Must there be a subspace of $V$ of codimension 1 disjoint from $A \cup B$?

**0**

votes

**1**answer

75 views

### Dimension of a similarity class

Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...

**1**

vote

**1**answer

101 views

### Invertibility of random Vandermonde matrix

Let $\kappa, d \in\mathbb{N}$ and $f$ is a uniform probability measure on $\mathcal{D} = \left[-1,1\right]^{\kappa}$. In addition, let
\begin{equation*}
p = p\left(\kappa,d\right) := ...

**5**

votes

**2**answers

190 views

### Local maxima and minima of the trace of a product of $SL_2^\pm(\mathbb{R})$-matrices

I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my ...

**3**

votes

**3**answers

575 views

### Algebraic K-theory can be seen as a generalization of Linear algebra? [closed]

Algebraic K-theory can be seen as a generalization of Linear algebra?
If yes, how so?

**3**

votes

**1**answer

366 views

### Is every element of $\mathrm{SL}(n,R)$ of finite order diagonalizable?

Let $k>0$ be an integer, let $R$ be a ring (commutative, unital), which contains $\mathbb{Q}$ (i.e. with a ring homomorphism $\mathbb{Q}\to R$) and all $k$-roots of unity. The examples I have in ...

**3**

votes

**0**answers

102 views

### Lattice with trivial spinor norm

Let $\Lambda$ be a non-degenerate lattice (over $\mathbb{Z}$) with quadratic form $q$. I define the spinor norm $\theta \colon O(\Lambda_\mathbb{R} )\to \lbrace\pm 1\rbrace$ as follows:
For a ...

**3**

votes

**4**answers

347 views

### Reference for an algebraic group preserving a cubic form

Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) ...

**2**

votes

**0**answers

318 views

### The twisted kiss of the curvaceous cubic and the staid tetrahedron (references)

(Migrated from MSE)
While investigating some operators, I came across some relations between the twisted cubic curve and the tetrahedron that link together some notions in differential geometry, ...

**3**

votes

**0**answers

204 views

### Find the Range of Function

What is the range of the map $\mathbb{C}^m\to\mathbb{C}^m$,
$$(z_1,\ldots,z_m)\mapsto (b_1,\ldots,b_m),$$
where $b_k=\prod_{j\neq k}(z_j-z_k)$ for $1\leq k\leq m$ ?

**9**

votes

**2**answers

475 views

### What are the invariants of $U\otimes V\otimes W$ under action of $GL(U)\times GL(V) \times GL(W)$

The tensor product of some (finite dimensional real) vector spaces is acted on by the direct product of their general linear groups. I would like to know if there are explicit invariants in the case ...

**6**

votes

**2**answers

294 views

### When are two subvarieties of matrices conjugate?

Let $X$ and $Y$ be two subvarieties of $n\times n$ matrices. My question is that is there any condition to guarantee that there exits some matrix $g$ such that $Y=g^{-1} X g$? If such $g$ exists, then ...

**4**

votes

**0**answers

151 views

### Dimension of a commuting nilpotent variety

Fix $k$ an algebraically closed field, $n$ a natural number, and $\lambda=(\lambda_1,\ldots,\lambda_m)$ a partition of $n$. Let $A$ be any $n\times n$ nilpotent matrix with entries in $k$ whose ...

**1**

vote

**0**answers

176 views

### An optimization problem on the sphere

Let $S$ be a sphere centered at origin in $\Bbb R^{2n}$ of radius $\sqrt{2n}$. Let $D$ be a diagonal matrix. Let $U$ be an orthogonal matrix. Let $r\in\Bbb Z_+$ be a fixed integer.
Let vector ...

**6**

votes

**1**answer

269 views

### Injectivity of matrix “fingerprint”

Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries.
For any matrix $A$, define
$$ ...

**0**

votes

**1**answer

144 views

### 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 ...

**2**

votes

**1**answer

200 views

### 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 ...

**0**

votes

**0**answers

120 views

### 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 ...

**0**

votes

**0**answers

60 views

### 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
...

**4**

votes

**2**answers

275 views

### Variety determined by interior product of the determinant?

Let $\Lambda^k(V)$ be the space of alternating $k$-linear tensors on $V$. Consider the map $f: \left(\mathbb{R}^n\right)^{n-k} \to \Lambda^k(\mathbb{R}^n)$ given by $\left(v_1,v_2, ..., ...

**0**

votes

**0**answers

182 views

### Monomial ideals: isomorphism problem for commutative algebras?

Theorem 5.27 in Polytopes, Rings, and K-Theory (Bruns, Gubeladze - 2009 - Springer SMM) claims:
Let $K$ be a field and $I\!\unlhd\!K[x]= K[x_1,\ldots,x_n]$ and ...

**4**

votes

**2**answers

286 views

### How to calculate the determinant bundle

Maybe, this is a problem of linear algebra. But I do not know how to calculate it. Let $E$ be a vector bundle of rank $2$ over an algebraic surface. If $H=S^{2n}E\bigotimes (\operatorname{det} ...

**2**

votes

**1**answer

141 views

### 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 ...

**0**

votes

**1**answer

190 views

### 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 ...

**2**

votes

**0**answers

200 views

### 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 ...

**7**

votes

**3**answers

627 views

### 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 ...

**2**

votes

**0**answers

111 views

### 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 ...

**1**

vote

**0**answers

75 views

### 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 ...

**2**

votes

**1**answer

153 views

### 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 ...

**1**

vote

**1**answer

153 views

### 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 ...

**10**

votes

**1**answer

379 views

### 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 ...

**3**

votes

**0**answers

87 views

### pavings and quadratic forms

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 ...

**0**

votes

**2**answers

163 views

### 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, ...

**4**

votes

**1**answer

232 views

### 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 ...

**7**

votes

**4**answers

634 views

### 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 ...

**0**

votes

**0**answers

156 views

### 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 ...

**2**

votes

**1**answer

171 views

### 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
...

**4**

votes

**2**answers

414 views

### 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 ...

**9**

votes

**2**answers

865 views

### 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, ...

**0**

votes

**1**answer

131 views

### 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 ...

**4**

votes

**2**answers

295 views

### 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 ...

**8**

votes

**4**answers

548 views

### 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 ...

**3**

votes

**2**answers

309 views

### 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 ...

**15**

votes

**3**answers

809 views

### 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 ...

**2**

votes

**0**answers

163 views

### 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 :
$E[X|X^Tb = c]$
...

**9**

votes

**1**answer

482 views

### Minimizing the excursion of a sum of unit vectors

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 ...

**4**

votes

**3**answers

322 views

### Linear subspaces in cones over orthogonal groups

Consider the orthogonal group $G=O(n)$ as a subset of the vector space of $n\times n$ real matrices. Let $C=C(G)$ denote the Euclidean cone over $G$, i.e., the space of matrices of the form $tA, A\in ...

**4**

votes

**4**answers

977 views

### The multiplicity of the max eigenvalue in matrix multiplication

Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq ...