All Questions
1,907 questions with no upvoted or accepted answers
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504
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How to compute the volume of a region transformed by a matrix?
This is a rewrite of the OP's question to emphasize what I think are the research level issues here.
Let $\mathscr{R}$ be a bounded convex body in $\mathbb{R}^n$ and let $H : \mathbb{R}^n \to \mathbb{...
- 550
5
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0
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436
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Non-diagonalizable positive matrices
Let $n\geq 3$ and $E_n$ be the set of $n\times n$ matrices $A$ satisfying the $3$ following properties:
$\bullet$ its entries $(a_{i,j})$ are positive integers.
$\bullet$ the eigenvalues of $A$ are ...
- 3,741
5
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91
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elementary matrices over a regular ring
Let $n\in\mathbb{N}$ with $n\geq 3$. Let $A$ be a regular ring and $\gamma\in GL_{n}(A)$. We suppose that $\gamma$ is homotopic to the identity, i.e. there exists $\alpha\in GL_{n}(A[X])$ such that $\...
- 3,472
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372
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Gelfand pairs in $SO(p,q)$
I am considering the groups $SO(p,q)$ over the reals. And inside it some parabolic subgroup, $P$. It can be the minimal parabolic but that is not the issue.
It is well known that $P$ has a Levi ...
- 51
5
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125
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Low-rank approximation over finite fields
Consider the finite field $\mathbb{F}_p$ with prime $p$. Let $I=\{ 0,1,2,...,|I|-1 \}\subset \mathbb{F}_p$ be an "interval".
What can be the largest size $|I|$, such that there exists a $2\times 2$ ...
- 51
5
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161
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Where have you encountered "arrangement spaces"?
I am compiling a paper in which I advertise (and use) the following notion of arrangement spaces (I made up the name, as I found no standard name in the literature).
Let $v_i\in\Bbb R^d,i\in N:=\{1,.....
- 13.6k
5
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350
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How to calculate the volume of a parallelepiped in a normed space?
Let $E$ be a real normed space, and let $v_1,...,v_n\in E$ be linearly independent. The parallelepiped defined by these vectors is $P=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Since $E$ is a ...
- 5,529
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150
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monomer-dimer tiling of a Young diagram
As a modest start, I propose the below problem for a special set of partitions. Perhaps it is known.
Let $\lambda_n=(n,n-1,\dots,2,1)$ be the staircase partition and its corresponding Young diagram $...
- 43.2k
5
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357
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$\text{Determinant}=(\sum \text{Determinant})^2$
Denote by $\delta_{n-1}=(n-1,n-2,\dots,1,0,0,\dots)$ the staircase partition and the embedded partition
$\lambda=(\lambda_1,\lambda_2,\dots)\subset\delta_{n-1}$.
QUESTION 1. Is this true?
$$\det\...
- 43.2k
5
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291
views
Can we represent partitions by mutually parallel lines in the plane?
Lately I have become interested in the following idea: Suppose $n$ is a positive integer and $[n]=\{1,2,3,...,n\}$. Suppose we have 3 distinct partitions $b$, $g$, and $r$ of $[n]$. Assume that the ...
- 1,277
5
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2k
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What is the Jarlskog invariant, conceptually?
Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity:
$$J_{ij,k\ell} := \operatorname{...
- 32.5k
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440
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Why does Faltings' Siegel lemma imply Siegel lemma?
Recall the Siegel lemma:
Let $A = (a_{ij})$ be an $N \times M$ matrix with rational integer coefficients. Put $a = \max_{i,j} |a_{ij}|$. Then, if $N < M$, the equation $Ax = 0$ has a solution $x\in\...
- 3,996
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397
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spectrum of orthogonality graphs
The orthogonality graph $\Omega(n)$ with $2^n$ vertices is the graph with vertex set $\{-1,+1\}^n$, with two vertices being adjacent if and only if they are orthogonal (as vectors in the standard ...
- 421
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162
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Homogeneous linear and quadratic inequalities
I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that
$$...
- 1,150
5
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856
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Dual of representation is irreducible implies the representation is irreducible?
Suppose $V$ is an infinite dimensional $\mathbb{Q}_p$-representation of Lie algebra $\mathfrak{g}$ over $\mathbb{Q}_p$. If its dual representation $V^{\prime}$ is irreducible then, is it always true ...
- 685
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586
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A minimal eigenvalue inequality
Suppose $A$ is an $n\times n$ real symmetric positive definite matrix. Let $A^{(-1)}_{i,j}$ be the $n\times n$ matrix the entries $(i,i),\,(i,j),\,(j,i),\,(j,j)$ of which equal to the corresponding ...
- 2,239
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a new proof of a square matrix is similar to its transpose (not completed)
I know there exist a classical way to prove that a square matrix is similar to its transpose in $C$ using jordan matrices.
I would like to prove it in another way but i have some difficulities to ...
- 159
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137
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How many CNOT gates are needed to compute scalar multiplication in a finite field of characteristic 2?
I want to know if scalar multiplication over a finite field of characteristic 2 is generally easier to compute than a linear transformation over a vector space over $F_{2}$.
Suppose that $p(x)$ is an ...
5
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answers
80
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Generalized rational form
Fix a field $k$. It is well known that any square matrix is similar to a block-diagonal matrix such that every block is of the following form: $1$ in the subdiagonal, a bunch of numbers in the last ...
- 59
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315
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Is there a matrix with this specific quadratic determinant?
We have $\det M=(a+b)(c+d)$ where
$M=\begin{bmatrix}
a& 0& -1& 0\\
0& c& 0& -1\\
b& 0& 1& 0\\
0& d& 0& 1
\end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
- 13.9k
5
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330
views
Best Approximation in Operator/non-Frobenius Norm
Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$
$$\...
- 153
5
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743
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Is this set empty?
Suppose we have two rank $n-1$ matrices in $\Bbb Z^{(n-1)\times n}$ given by
$$C=\begin{bmatrix}
c_{1}& -d_{1}& 0& 0&\dots 0& 0\\
0& c_{2}& -d_{2}& 0&...
- 13.9k
5
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235
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Riemann theta function inequality for a class of large random matrices
The following is essentially the same question as in this previous post, but since I have completely re-formulated it (hopefully for the better ;-), I decided to post a new question instead of an edit....
- 686
5
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240
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Complexity of approximating the size of the range of a matrix
Given an $m$ by $n$ matrix $M$ with $m \leq n$ and elements from $\{-1,1\}$, let us define:
$$S_M = |\{Mx : x \in \{-1,1\}^n\}.$$
It is NP-hard to compute $S_M$ exactly I believe by applying the ...
- 3,377
5
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234
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A combinatorial problem
What is the largest $m\times m$ $0/1$ matrix of real rank $n$ with every square submatrix sized at least ${n^{\gamma}}\times{n^{\gamma}}$ distinct for some fixed $\gamma>0$?
Upper Bounds: Number ...
user76479
5
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216
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Existence or construction of a sequence of orthogonal matrices with three properties
This is a problem that I encountered during my research, and I have spent a good amount of time on it without success. So I am reaching out for help ....
Any pointers or suggestions are appreicated!
...
- 984
5
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251
views
Eigenvalues of a certain product of matrices with special structure
Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = [G\hspace{1em}\...
- 6,853
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135
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Relative invariants of $P\oplus P^*$
Let $P$ be a $\mathrm{GL}(V)$-module, and assume that the decomposition of $P$ into irreducible submodules is known. By a relative invariant of the module $P\oplus P^*$, I mean a homogeneous nonzero ...
- 2,527
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310
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Biggest (or large) rectangle in a polytope
I need an efficient method to construct a (hyper)rectangle inside a polytope with a lot of dimensions (say $100 < d < 1000$). Ideally I'd want the biggest possible rectangle, but as I don't ...
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187
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Mapping the standard $(n!-1)$-simplex into $GL_n(\mathbb C)$
Let $\Delta^{n!-1}$ be the standard $(n!-1)$-simplex whose vertices are indexed by the elements of the symmetric group $S_n$, that is
\begin{equation}
\Delta^{n!-1} = \left\{ (t_\sigma)_{\sigma\in S_n}...
- 3,984
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376
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Non-linear positive map
In the paper titled "Nonlinear completely positive maps" M. D. Choi and T. Ando extended natural definition of completely positive maps ignoring the linearity condition (Aspects of positivity in ...
- 421
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254
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A weak Perron-Frobenius property for sets of positive matrices
A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...
- 6,206
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620
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Is there a method to simultaneously block-diagonalize a set of group matrices?
Assume that you are explicitly given the representation matrices of a group.
How does one go about finding that common basis which will find the irreducible components of all of them simultaneously?
...
- 1,893
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answers
167
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A specific case of the $p$-center problem
Given a fixed positive integer $m$, let $\cal{S}$ be the subset from $\mathbb{R}^m$ defined as $\cal{S} = \{u \in \mathbb{R}^m \mid \forall i \in \{1, \dots, m\}, u(i) > 0$ and $\sum_{i=1}^m{u(i) = ...
- 125
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255
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Existence of a matrix product from its eigenvalues
Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia):
$$
\lambda^\downarrow(A) \cdot \lambda^\...
- 151
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428
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Weyl's construction for symplectic groups--an exercise in Fulton and Harris's book
This is an exercise in section 17.3 in Fulton and Harris's book:Representation theory-a first course.
Let $V=\mathbb{C}^{2n}$ and $Sp(2n)$ be the symplectic group w.r.t the nondegenerate bilinear ...
- 181
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A stronger Cauchy-Schwarz inequality for traces of compression matrices
Assume that $A$ and $B$ are contractions, so
$I-AA^T$ and $I-BB^T$ are positive-definite matrices. Let
$C=(I-AB^T)^{-1}(I-AA^{T})(I-BA^{T})^{-1}$, and show that:
$$Tr\left(\frac{1}{1-AA^T}\right)...
- 4,270
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235
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A question on hyperplanes in partial linear spaces and hypergraphs
A partial linear space (or a linear hypergraph) is a point line geometry $(P,L,I)$ where for every pair of points there is at most one line incident with both of them. A hyperplane in a partial linear ...
- 1,197
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604
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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, ...
- 10.5k
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148
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Groups of operators between local unitaries and full unitaries
Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
- 6,351
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194
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A linear optimization problem on a graph
Let $G=(V,E)$ be a finite graph and let $f$ be any positive function defined on the vertices. Put weights on the vertices $v_{i}$, way $w_{i}$ so that $\sum_{i=1}^{n}w_{i}\leq 1$. Assume that every ...
- 2,288
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231
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Singularity of an $l\times l$ matrix whose entries are $2l$-th roots of unity
Let $l$ be a positive integer, $\zeta$ be a primitive $2l$-th root of unity in $\mathbb{C}$, and $\alpha,\beta$ be $\pm1$ sequences of length $l$, i.e. $\alpha_k=\pm1,\beta_k=\pm1$ for $k=0,\dots,l-1$....
- 767
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392
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Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix
I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D -...
- 500
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0
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112
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Inverses of the sums of all possible subsets of a set of symmetric and positive definite matrices
I have a set of $c$ matrices $A_1 ... A_c$ which are all symmetric and positive definite. I would like to calculate the inverses of all the possible sums, i.e.
$(A_1+A_2)^{-1},(A_1+A_3)^{-1},(A_2+A_3)...
- 71
5
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answers
153
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What is the composition in SesquiAlg?
To motivate my question, I will describe a famous 2-category. First, there is the 1-category $\text{Vect}$ of vector spaces (over some fixed field). This category has a tensor product $\otimes$, and ...
- 54.6k
5
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406
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On the linear transformation between matrix space
Suppose we have two matrix subspaces, $n\times n$ matrix subspace $S_1$ and $m\times m$ matrix subspace $S_2$. Every element of $S_1$ and $S_2$ is complex symmetric matrix.
Suppose there exists ...
- 1,503
5
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0
answers
1k
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Tensors as multilinear maps
I am aware that many books on differential geometry define tensors as multilinear maps. Namely
$$
V\otimes W := L_2(V^* \times W^*,\Bbb F)
$$
I am also aware that this space is isomorphic to the ...
- 61
5
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0
answers
596
views
Literature on Exponential of a Quadratic Form
Let $A_i$, $i=1,\dots,L$ be given $N\times N$ positive definite real matrices. I have this sum of exponentials
\begin{align}
f(\mathbf{x})=\sum_{i=1}^{L}\operatorname{exp}(-{\mathbf{x}^T\mathbf{A}_i\...
- 1,421
5
votes
0
answers
442
views
A reference on semisimple linear algebra
Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...
- 4,000
5
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0
answers
160
views
reference for perturbation of projection result
Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...
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