2
votes
0answers
192 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
2answers
324 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
2answers
287 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
0answers
139 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
0answers
163 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
1answer
240 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
1answer
128 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
1answer
188 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
0answers
109 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
0answers
55 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 ...
2
votes
1answer
104 views

Union of orthogonal complements of subspaces is not contained in a proper algebraic variety

Consider an $n \times n$-matrix $A$ and an $m \times n$-matrix $C$ where $m < n$. For each $t \ge 0$ the kernel $\ker Ce^{At}$ is a (say $k$-dimensional) subspace. Suppose the intersection of these ...
3
votes
2answers
231 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
0answers
163 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 ...
5
votes
2answers
282 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} ...
1
vote
1answer
111 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
1answer
182 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
0answers
188 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 ...
5
votes
2answers
400 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 ...
1
vote
0answers
105 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
0answers
72 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
1answer
147 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
1answer
144 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
1answer
362 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
0answers
85 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
0answers
222 views

can I find the rotation matrix R and translation matrix T from 3x3 matrix?

In the pinhole camera model, I can get the homography 3x3 matrix of two images. My problems is: provided an camera intrinsic matrix(the projection matrix), can I find the find the rotation matrix R ...
0
votes
0answers
32 views

Last Point of Exit from the 2-D Positive Quadrant

I define the functions $f_i(\mathbf{u}),i=1,2$ and $g_i(\mathbf{u}),i=1,2$ where $\mathbf{u}\in\mathbb{C}^{N}$ is the unit-norm vector. Thus, this functions are defined over the unit norm $N-$sphere. ...
0
votes
2answers
162 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
1answer
210 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 ...
6
votes
4answers
574 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
0answers
142 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
1answer
162 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
2answers
351 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 ...
8
votes
2answers
689 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
1answer
129 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
2answers
293 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
4answers
519 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
2answers
290 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
3answers
764 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
0answers
158 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
1answer
462 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
3answers
317 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 ...
3
votes
3answers
865 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 ...
3
votes
1answer
667 views

Closest 3D rotation matrix in the Frobenius norm sense

Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm: \begin{equation} \|R-M\|_F \end{equation} Is there a closed form solution for $R$, or is it ...
8
votes
0answers
498 views

An elementary linear algebra problem

Let $K$ be a field, and let $E$ be the algebra of $n\times n$ matrices over $K$. Let $V_0$ and $V_1$ be the (left) $E$-modules of matrices of size $n\times n_0$ and $n\times n_1$. Let $W \subseteq ...
6
votes
1answer
1k views

Constructing a unitary matrix

Setting: Given a set of $n\times n$ matrices $A_i$, I would like to find a linear combination of these matrices $Q = \sum_i A_i x_i$ with $x_i$ a set of complex numbers, such that $Q$ is unitary: ...
5
votes
3answers
700 views

Finding the action of the symplectic group on the Siegel-half plane

Let $S$ be the Siegel-half plane of dimension $n$, i.e. the set of complex $n \times n$ matrices $Z$ which are symmetric and whose imaginary part is positive-definite. In dimension 1 we can identify ...
3
votes
1answer
315 views

Mathematical Programming with other Algebras than Linear

Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization. What analogies are there for ...
3
votes
3answers
382 views

Multiplicity of eigenvalues in 2-dim families of symmetric matrices

Say you have 2 symmetric matrices, $A$ and $B$, and you know that every linear combination $xA+yB$ ($x,\\,y\in \mathbb{R}$) has an eigenvalue of multiplicity at least $m>1$. Such a situation can of ...
2
votes
0answers
495 views

Ideal membership

Let $n=2t$ be an even number. Let $F$ denote a finite field where $|F|=q$. Let $A_{1}, A_{2},\ldots, A_{t}$ and $B_{1},B_{2},\ldots,B_{t}$ be distinct matrices in $M_{n}(F)$. Let $$ X = ...
3
votes
2answers
466 views

$k$ structures on $K$ vector spaces

The following statement is made in Borel's Linear Algebraic groups, section 11 on $k$ structures. Let $V$ and $W$ be $K$ vector spaces with $k$ structures. If $f:V\to W$ is $K$ linear, then $f$ is ...