All Questions
921 questions with no upvoted or accepted answers
4
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112
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Inducing surjections on $GL_n(-)$?
Suppose $A,\,B$ are (possibly noncommutative) rings, and $GL_n(-)$ is the group of invertible $n\times n$ matrices over a given ring. Suppose $f:A\to B$ is surjective, does it necessarily follow that $...
- 61
4
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0
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224
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Equations in finite subgroups of unitary groups
Let $n$ be an integer. Andreas Thom mentioned that Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ ...
- 24.8k
4
votes
0
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437
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A sum of Ramanujan sums
I have the following question about Ramanujan sums.
(All vectors and matrices here will be understood to have integer entries.)
Let $X_q=\{ (x_1,...,x_R)|1\leq x_i\leq q\} $ and let, for any $R\...
- 1,381
4
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0
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163
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Matrices in $SL(2,\mathbb{C})$ with characteristic polynomial defined over a subring
Let $R\subset\mathbb{C}$ be a subring, and let $A,B\in SL(2,\mathbb{C})$ be matrices such that $A,B,AB$ all have trace in $R$.
For which $R$ can we then deduce that $A,B$ are simultaneously conjugate ...
- 7,527
4
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0
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84
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Matrices with almost constant coefficient have a simple eigenvalue
As a by-product of a general result for bounded operators of a Banach space, I have the following:
A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
- 14.4k
4
votes
0
answers
168
views
Fast matrix-vector product for structured matrices
Let $X\in\mathbb{C}^{m\times n}$ be a matrix that satisfies the Sylvester equation
$$AX-XB = F,\qquad A\in\mathbb{C}^{m\times m}, \quad B\in\mathbb{C}^{n\times n},$$
where $F\in\mathbb{C}^{m\times n}$...
- 41
4
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374
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non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?
We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e., the matrix is not only weak diagonal-dominant, but ...
- 329
4
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0
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208
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An operator derived from the divided difference operator $\partial_{w_0}$
Some main definitions and basic facts of divided differences:
In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...
- 1,997
4
votes
0
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125
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How to enumerate a discrete group of matrices by their Frobenius norm?
Suppose I have a discrete group $G<\mathrm{SL}_2(\mathbb{C})$,
and it is finitely generated by some known generators.
That is, $G=\langle g_1,\dots,g_n\rangle$.
The Frobenius norm of a matrix $m=\...
- 2,436
4
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0
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514
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Efficiently calculate the trace of the product of two large but symmetric matrices, one of which is an inverse
Sorry about the long title. I need to calculate the trace of $M(M+D)^{-1}$, where $M$ is a dense symmetric matrix, and $D$ is a diagonal matrix. The main issue is the dimension could be large (usually ...
- 41
4
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0
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431
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Spectral radius of the product of a right stochastic matrix and a block diagonal matrix
Let us define the following matrix:
$C=AB$
where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$ … $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M - \mu R_k$ with $R_k$ ...
- 41
4
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0
answers
676
views
Weyl-type inequality for non-Hermitian matrices?
What is the weakest known condition under which a Weyl-type eigenvalue perturbation inequality holds? Does some analogue hold for normal matrices, for example?
- 6,541
4
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0
answers
171
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How does scaling rows to sum to 1, of a positive matrix change the perron vector?
Reposting from math.sx due to lack of response. Let $A$ be a $N\times N$ positive matrix such that $A_{ij}>0$. By Perron-Frobenius theorem, there is a unique positive left eigenvector called Perron ...
- 1,421
4
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0
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234
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Relation between Euclidean distance matrices and squared-distance matrices
Let us refer to $D$ as the matrix with $D_{ij} = ||x_i - x_j||_2$ and $H = D \circ D$ e.g. the matrix with $H_{ij} = ||x_i - x_j||_2^2$ as its entries. Are the spectra of these matrices related at all?...
- 133
4
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89
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Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?
Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries).
Can we partition this union into at most $n$ rectangles?
I think it's pretty ...
- 18.7k
4
votes
0
answers
342
views
Determinant of the sum of a psd (Kronecker) matrix and a diagonal matrix?
Let $K = K1 \otimes K2$ where $K1$ and $K2$ are positive semidefinite matrices. Let $W$ be a diagonal matrix with positive entries. (Everything is real-valued.)
I want to calculate or bound $\det (...
- 141
4
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answers
164
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Counting cosets of matrices of determinant > 1 under the action of a congruence subgroup
I tried asking this on math exchange, but no luck, so thought I'd try here.
Let $M_2(m,\mathbb{Z}) $ be the $2\times 2$ matrices with integer entries and determinant $m$. Let $\Gamma^0(N)$ be the ...
- 41
4
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1k
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How to find the unitary matrices in this exponential matrix representation
In the following post
Representing a product of matrix exponentials as the exponential of a sum
there is a statement regarding the result of the multiplication of two matrix exponentials:
if $A$ and $...
- 149
4
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0
answers
217
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Examples of functions from matrices to real numbers with certain properties
Let $M(\mathbb{R})$ be the set of all matrices (of any size) over $\mathbb{R}$. Let $v : M(\mathbb{R}) \rightarrow \mathbb{R}$ be a function which satisfies the following 5 properties:
If $\mathbf{a}...
- 101
4
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0
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657
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determinant of fibonacci-sum graphs
We have a simple graph with vertices $\{v_1, v_2, ... v_n\}$.
The adjacency matrix of this graph is $A= (a_{ij})$ so that
$a_{ij}=1$ if $i+j$ belongs to the Fibonacci sequence;
$a_{ij}=0$ ...
- 41
4
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0
answers
154
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connectivity in automata by words of length n-1
Let $A$ be a complete strongly connected automaton with $n$ states. Does always exist a word $v$ of length at most $n-1$ such that its underlying graph is connected?
That is for any pair of distinct ...
4
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0
answers
1k
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Cartan decomposition for upper triangular matrices
Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$.
Let $K= GL_n(o)$ and let $I$ the Iwahori ...
- 11.2k
4
votes
0
answers
434
views
Smallest matrix covered by many random n by n matrices
We say that a matrix $M$ can be covered by a (smaller) matrix $N$ if every entry in $M$ is contained in some submatrix of $M$ that exactly equals to $N$, up to reordering the rows and columns of $N$. ...
- 51
4
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answers
3k
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The determinant of the hadamard product of two matrices
We know that the determinant of a Hadamard product of two positive semidefinite matrices $|{\bf A}\circ{\bf B}|$ is greater than or equal to $|{\bf A}||{\bf B}|$. Are there any general results on ...
- 449
4
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261
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Growth of symmetric positive definite integral matrices.
Given an integer $d$, let $\alpha_d(N)$ denote the number of symmetric integral positive definite matrices of size $d\times d$ with coefficients in $\lbrace -N,-N+1,\dots,N-1,N\rbrace$.
...
- 17.9k
4
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256
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A matrix minimisation problem
Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\...
- 18.7k
4
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0
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453
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Convergence of the relaxation method for every parameter in the relevant disk
For large size matrices, the resolution of linear systems $Ax=b$ is often done iteratively. The matrix $A$ is split as $A=M-N$, with $M$ invertible, and one performs
$$x^{k+1}=M^{-1}(Nx^k+b).$$
The ...
- 52.3k
4
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0
answers
562
views
The order of the Jacobi method for Hermitian matrices
Let $H$ be an $n\times n$ Hermitian matrix. The Jacobi method is an iterative method for finding the spectrum of $H$. It is described in every book on numerical linear algebra.
Principle: At step $k$,...
- 52.3k
4
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0
answers
351
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Combinatorics of signed oriented graphs/skew-symmetric matrices
Consider a "complete" signed graph on $n$ vertices indexed by
$1,2,\dots,n$, that is, a graph in which any two distinct vertices $i$ and $j$ are connected by an oriented edge.
For each pair of ...
- 13.5k
4
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0
answers
733
views
Haar measure on strictly upper triangular matrices
Let F be a function field, and A its adele ring. I want to consider U(A)/U(F), where U(A) is the space of strictly upper triangular matrices with entries from A, and U(F) is the same with entries ...
- 41
3
votes
0
answers
118
views
A matrix-valued analogue of a classical inequality
Let $p \geq 4$ be an even integer. In the study of variational problems in $W^{1, p}$, it is handy to know that for $a, b \in \mathbb R^d$,
$$|a - b|^p \leq 2^{p - 1} (|a|^{p - 2} + |b|^{p - 2}) |a - ...
- 708
3
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0
answers
109
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How much a general a theory of matrices equivalence under group actions we have?
Let $F$ be a field and let $M_{m,n}\,(F)$ be the $F$-linear space of $m \times n$ matrices over $F$. Let $G$ be a group acting on $M_{m,n}\,(F)$.
My question is: Do we have some theory about the ...
- 157
3
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0
answers
58
views
About a circular variant of Vandermonde matrix
Given an arbitrary $(x_1, \dots, x_n) \in [0, 1]^n$, is there any name/known results for the following $n \times n$ matrix (which is constructed by iterating $(x_1 \to \dots \to x_n \to x_1 \to \dots)$...
- 33
3
votes
0
answers
83
views
A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal
I apologize if this is too elementary a question, but I have not been able to make much progress.
Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
- 251
3
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0
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452
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Dimension of a subspace of $n\times n$ real symmetric matrices
Let $n\in \mathbb N.$ Let $W$ be a non-trivial subspace of $n\times n$ symmetric matrices such that for every $x\in \mathbb R^n\setminus \{0\}$ there exists $a_x\in \mathbb R^n\setminus \{0\}$ such ...
- 49
3
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0
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145
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Eigenvalues of random matrices are measurable functions
I have read that if a random matrix is hermitian then its eigenvalues are continuous, hence also measurable.
If the random matrix is not hermitian, the eigenvalues are not continuous in some cases. ...
- 31
3
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0
answers
83
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How many local maxima can $(x_1,\dots,x_r)\mapsto\|x_1A_1+\dots+x_rA_r\|_\infty/\|(x_1,\dots,x_r)\|_2$ have for Hermitian $A_1,\dots,A_r$?
Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Suppose that $A_1,\dots,A_r\in M_n(K)$ are all Hermitian.
Define a function $f_{A_1,\dots,A_r}:\mathbb{RP}^{n-1}\rightarrow[0,\infty)$ by setting
$$f_{...
3
votes
0
answers
130
views
The probability that the dominant eigenvalue of a random real matrix is real
Let $X_n$ be an $n\times n$ real matrix where the entries in $X_n$ are independent, normally distributed, have mean $0$, and variance $1$. Suppose that $\lambda_1,\dots,\lambda_n$ are the eigenvalues ...
3
votes
0
answers
85
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Exterior powers of the Cartan matrix and Dyck paths
(This question can be formulated purely combinatorially in terms of Dyck paths, which is done in the second part of the question. But I am more interested whether this can be explained by some sort of ...
- 26.5k
3
votes
0
answers
250
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Generalized matrix determinant lemma for pseudo-determinant of symmetric matrix
The pseudo-determinant of a square matrix $A$ is the product of its nonzero eigenvalues. Consider the generalized matrix determinant lemma $$\det(A+UWV^\top) = \det A\det W\det(W^{-1} + V^\top A^{-1}U)...
- 141
3
votes
0
answers
142
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Solvability of a matrix exponential equation - generalized matrix logarithm
For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation
$$
G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) .
$$
Basic ...
- 1,081
3
votes
0
answers
295
views
Decomposition of a determinant
Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$.
Does there exist a ...
- 8,998
3
votes
0
answers
163
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On a decomposition of $2\times2$ integer matrix
Given an integer matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ with determinant $det$ and two positive integers $m,n:m+n=det$, when is it possible to write it sum of two integer matrices $\...
- 13.9k
3
votes
0
answers
43
views
Different generating sets for conjugation invariants of several matrices
There is a theorem of Procesi that the ring of polynomial functions on tuples $(A_1,A_2, \dots, A_m)$ of $n \times n$ matrices, which are invariant under simultaneous conjugation, is generated by ...
- 213
3
votes
0
answers
132
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the growth rate of poly-$\mathbb{Z}$ group
I am interested in the growth rate of the poly-$\mathbb{Z}$ group. Let $G$ be a poly-$\mathbb{Z}$ group, i.e $$G =(\dots((\mathbb{Z} \rtimes_{\phi_1} \mathbb{Z})\rtimes_{\phi_2} \mathbb{Z}) \rtimes_{\...
- 823
3
votes
0
answers
538
views
Diagonalizing a block tridiagonal matrix
Is there an efficient way to diagonalize a block tridiagonal $N\times N$ matrix of the following form:
\begin{matrix}
A_0 & B & 0 & 0 & \ldots \\
B & A_1 & B & 0 & \...
- 31
3
votes
0
answers
211
views
Conjugacy class of upper triangular matrices over algebraically closed field: Reference request
We know that the conjugacy classes of $A\in M_n(\mathbb{C})$ are determined by the characteristic polynomial of $A$ and a partition of $n$. Is there an analogous statement for upper triangular ...
- 265
3
votes
0
answers
40
views
Invertibility of the sampling matrix
Given a function $f: \mathbb{R}^2\rightarrow\mathbb{C}$ sampled as a matrix $F_{ij}$ on some ractangle $[a,b]\times[c,d]\subset\mathbb{R}^2$ with steps $\Delta x$ and $\Delta y$ as the stepsizes so ...
- 61
3
votes
1
answer
290
views
Cofactor matrices and positive semi-definiteness
If $A$ is an $n\times n$ matrix with real entries, let me write $\widehat A$ its cofactor matrix. Since the map $A\mapsto\widehat A$ is polynomial, homogeneous of degree $n-1$, it can be multi-...
- 52.3k
3
votes
0
answers
147
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Convolution integral and its matrix representation
My background is chemistry and I was exploring some one dimensional deconvolution problems i.e., resolution of two or more overlapping peaks. A lot of excellent work was done in the 1970-80s. However, ...
- 879