All Questions
2,026 questions with no upvoted or accepted answers
1
vote
0
answers
260
views
Product of exponentials of matrices
Let $A$ and $B$ be commuting $n\times n$ positive definite complex matrices and let $C$,$D$ be other complex matrices. I wish to think of $C,D$ as small and $A,B$ as any size.
Suppose we are looking ...
- 280
1
vote
0
answers
63
views
Does the equation $\mathbf{I} = \sum_{k=1}^{m} \mathbf{U} \mathbf{B}_k \mathbf{U}^T\mathbf{A}_k$ have a special name or solution?
I have encountered the equation $\mathbf{I} = \sum_{k=1}^{m} \mathbf{X} \mathbf{B}_k \mathbf{X}^T\mathbf{A}_k$, recently.
All matrices are of dimension $n \times n$.
Is it assigned a special name?
...
- 39
1
vote
0
answers
127
views
Low-rank approximation of sub-sampled matrix
Considering a large data matrix $X$ with zero-centered columns that is assumed to be approximately low-rank, it is common to do PCA and project the data onto the top few principal components, and use ...
- 111
1
vote
0
answers
229
views
On Hochschild cohomology
Let given a ring $A$ without zero divizors and torsion free as $\mathbb{Z}$- module. Let also given $M~-$ $A~-$ bimodule which is torsion as module over $\mathbb{Z}$.
Is it true that if $A$ has ...
- 291
1
vote
0
answers
460
views
Orthonormal basis of matrices
I am asking if somebody knows how to do or is aware of the following construction:
Let $n \in \mathbb{N}$ be given, then take an arbitrary matrix $A \in \mathbb{C}^{n \times n}$. Then, there are maps ...
- 21
1
vote
0
answers
81
views
Classification of irreducible complex representations of $\mathbb{H}_\mathbb{Z}^\times$ up to isomophism?
Let $G$ be the multiplicative group of the ring of integral quaternions.
Question. What is the classification of irreducible complex representations of $G$ up to isomorphism?
user74319
1
vote
0
answers
79
views
Evaluating a hopeless algorithm for recovering sparse solutions to $Ax=b$ over a finite field
Given a matrix $A \in F^{n \times m},$ $m>n,$ with rank $n$ and in row-reduced echelon form and a nonzero vector $b \in F^n$: (1) Perform elementary row operations to obtain $b_1 \not = 0$ and $b_k=...
- 259
1
vote
0
answers
398
views
Center of matrices
I encountered a neat problem in a problem in particle physics
So given $n$ skew symmetric matrices $A_1,...,A_n$ in $\mathbb{C}^{d \times d}.$
I would like to call this the commutator property: $...
1
vote
0
answers
256
views
Significance of Tikhonov matrix
I am looking for a tutorial on Tikhonov matrix, in the sense what it can do or it cannot do. The definition of the matrix can be obtained in the wikipedia link. https://en.wikipedia.org/wiki/...
1
vote
0
answers
264
views
Maximizing the ratio of largest eigenvalues
Let $K$ be a real stable matrix; more specifically,
$$
K=\left(\begin{array}{rrrrr}
0&1&0&\ldots&0\\
0&0&1&\ldots&0\\
\vdots&\vdots&\vdots&\ddots&\vdots\...
- 270
1
vote
0
answers
154
views
When can a binary matrix be transformed into a certain form
I have a $k \times n$ matrix $G$ over ${\mathbb F_2}$ that's full rank.
This can always be put in systematic form : $G \sim [I_k \mid P]$ where $I_k$ is a $k \times k$ identity matrix and $P$ is a $k \...
- 451
1
vote
0
answers
78
views
Does this graph structure on a ring have any known uses, and/or is it studied anywhere?
Every ring $R$ becomes a graph as follows: given $a,b \in R$, an edge between $a$ and $b$ is, by definition, a non-zero idempotent of $R$ that when added to $a$ yields $b$. This is always a simple ...
- 3,793
1
vote
0
answers
55
views
On the Lowener-Heinz inequality
I know that for two symmetric positive semi-definite (non-diagonal) matrices $A,B$, the inequality asserts that the following does not hold for all $p > 1$
$$A \succeq B \succeq 0 \Rightarrow A^p \...
- 11
1
vote
0
answers
39
views
Quantifying how much a vector gets turned toward the expanding direction of an $\mathrm{SL}(2,\mathbb R)$ matrix
Consider a matrix $B\in \mathrm{SL}(2,\mathbb R)$. Let $s$ be a vector that is pointing in the most contracted direction of $B$, and let $u$ be the image under $B$ of a unit vector pointing in the ...
- 785
1
vote
0
answers
91
views
determine the existence of positive semi-definite matrix
Given a $d\times d$ complex matrix subspace $S$, we are asking whether there is some finite integer $n$ such that there exists a non-zero positive semi-definite matrix is orthogonal to $S^{\otimes n}$....
- 1,503
1
vote
0
answers
72
views
4-D lattices and quaternion
It is easy to prove that there are only 2 extensions $\mathbb{Q}(a)$, with $|a|=1$, of $\mathbb{Q}$ where $\mathbb{Z}[a]$ becomes a lattice(discrete free abelian subgroup of rank 2) in the complex ...
- 11
1
vote
0
answers
111
views
Number of Asymmetric, Balanced Permutation Matrices
let $\mathcal{ABP}$, the set of "asymmetric, balanced permutation matrices", be defined as the set of permutation matrices, that aren't equal their transpose but, for which the number $1$s below the ...
- 13.2k
1
vote
0
answers
328
views
Binary operations on graphs
Are there "binary operations" on graphs like in (https://en.wikipedia.org/wiki/Graph_product), which make the set of all graphs ("under consideration")
a (abelian) group or
a (commutative) ring or
a ...
user6671
1
vote
0
answers
68
views
Computing intersection of Weyl algebra ideal with certain subring
Let $D=k [x_1,\ldots, x_n, \partial_1,\ldots, \partial_n] $ be the nth Weyl algebra over the characteristic zero field $k $. Set $\theta_i=x_i\partial_i $. Let $I $ be a left ideal in $D $. Is there a ...
- 3,079
1
vote
0
answers
174
views
Negative eigenvalue of Toeplitz Hermitian matrix?
I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
- 495
1
vote
0
answers
241
views
when a prime ideal is maximal differential ideal in a UFD
Is the prime ideal $\langle X^{2}+Y^{2}-1\rangle$ a maximal differential ideal in differential ring $\mathbb{Q}[X,Y]$ with
derivatives $D(X)=Y, D(Y)= -X$?
I know there are maximal ideals like $\...
- 11
1
vote
0
answers
269
views
Transitivity for algebraic extensions of integral domains?
I'm trying to prove the following result. Let $R_1\subseteq R_2\subseteq R_3$ be integral domains such that $R_1\subseteq R_2$ and $R_2\subseteq R_3$ are algebraic extensions (note: I do not want to ...
- 3,761
1
vote
0
answers
226
views
The projective resolution of a direct summand
For an $R$-module $M$ fix a projective resolution $P^\bullet\to M$. If $N$ is a direct summand of $M$, that is there is $L$ such that $M=N\oplus L$, then is there a projective resolution of $N$ which ...
- 666
1
vote
0
answers
118
views
Asymptotic determinant of $2\times 2$ Toeplitz matrix
The problem that I am dealing with is to compute the determinant of a $2\times 2$ Toeplitz matrix[1] (in general I would like to generalize to a more general case, but let's consider the easiest case ...
- 183
1
vote
0
answers
517
views
Complex conjugate and unitary complex conjugate
Definition: Let V be complex finite dimensional inner product space
Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$
Definition: Anti-unitary Complex ...
- 121
1
vote
0
answers
98
views
Roots of matrices in $G_2(Z)$
Let $G_2$ denote the exceptional Lie group $G_2$ as a $\mathbb{Q}$-algebraic group. Suppose that is also given a matrix representation $\rho : G_2\rightarrow SO(7)$. Let $M$ be a matrix with integral ...
- 11
1
vote
0
answers
137
views
Boundary of pseudospectra
Suppose:
$B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$
${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
- 151
1
vote
0
answers
97
views
A property of minimal prime ideals in rings with finite chromatic number
Let $R$ be a commutative ring with identity. There are so many ways to associate a graph to $R$. Consider this: take the elements of $R$ (All elements including zero) as vertices an two distinct ...
- 271
1
vote
0
answers
207
views
the linear span of all matrix coefficients is $C(G,\mathbb{C})$ where $G$ is a finite group
Theorem. Let $\{(R_{\alpha},V_{\alpha})\}$ be a complete set of inequivalent irreducible finite dimensional representations of a finite group $G$. Let $V_{R_{\alpha}}$ be the subspace generated by all ...
- 155
1
vote
0
answers
105
views
Primitivity of $AA^\top$
Let $A\in\mathbb{R}^{n\times n}$ be a non-negative and irreducible matrix. Consider $B:=AA^\top$. It can be proved (I can post a proof if needed) that the following condition is necessary and ...
- 2,712
1
vote
0
answers
96
views
Let $M = \frac{1}{2}(A + {A^T})$ be real symmetric nonnegative matrix. Why does $\rho (A) \le {\lambda _{\max }}(M)$?
Let $A \in M_n$ be nonnegative, and consider the real symmetric nonnegative matrix
$M = \frac{1}{2}(A + {A^T})$.
Why does $\rho (A) \le {\lambda _{\max }}(M)$?
- 11
1
vote
0
answers
26
views
How can I filter the effects of a variable from a correlation matrix?
I have a correlation matrix (it contains 500 columns and 500 rows) and I would like to make an other correlation matrix in which one variable (and its influences) is filtered from the initial matrix. ...
- 11
1
vote
0
answers
214
views
Dimension of the center of a subalgebra of a simple algebra
Let $F$ be a field. Let $A$ be a simple (associative unital) $F$-algebra with center reduced to $F$. Let $B$ be a $F$-subalgebra of $A$;
assume that $A$ is can be generated as left $B$-module by $n$ ...
- 11
1
vote
0
answers
94
views
Keshavan-Montanari-Oh matrix completion — clearing step
I am trying to implement the algorithm for matrix completion proposed by Keshavan, Montanari and Oh (2009). It consists of three steps:
Trimming which nulls some rows and columns to make the high ...
- 31
1
vote
0
answers
206
views
The normalizer problem for group rings
I recently studied about The Normalizer problem (NP) which states that given an integral group ring $\Bbb{Z}G$, $N_{\cal{U}}(G)=G\frak{z}$ where $\frak{z}$ denotes centre of $\cal{U}$ = $\cal{U}$$(\...
1
vote
0
answers
282
views
Expected value of minimum rank of random matrices
I have $n$ random vectors ${\bf r}_i$ for $i=1,2,\dots,n$, each with dimension $1 \times m$, and $n$ random matrices ${\bf S}_i$ for $i=1,2,\dots,n$, each with dimension $M \times m$. The elements of $...
- 307
1
vote
0
answers
131
views
Lower bound on the value $\textbf{1}^Tx$ such as $Ax\geq b$
The problem may be formulated as follows:
We are given a set of $m$ positive numbers $\{b_1,...,b_m\}$ and a set of $n$ positive numbers $\{v_1,...,v_n\}$. We have $v_j\leq K$, $j=1,...,n$, for a ...
- 123
1
vote
0
answers
167
views
Embedding of fields in central simple algebras over number fields
Let $K$ be a non-real CM number field of degree $2d$, with maximal totally real subfield $K_0$, and let $A$ be a central simple algebra over $K$, so that $A\simeq M_n(E)$, the $n\times n$ matrix ring ...
- 2,406
1
vote
0
answers
35
views
Projectivity of a faithfully balanced self-orthogonal bimodule
Let $_RT_S$ be a faithully balanced self-orthogonal bimodule over a pair of noncommutative rings $(R,S)$, if $_RT$ is projective as a left $R$-module, can we say $T_S$ is also projective as a right $S$...
- 1,207
1
vote
0
answers
141
views
For of a special case of $Ax>=b$, are there always integer solutions?
The input consists of two integers $n\geq 2$ and $K\geq 2$, and a vector of positive integers $b$ of size $n$. We assume that $\sum_i b_i$ is a multiple of $K$.
The output is a matrix $A(n\times n)$ ...
- 123
1
vote
0
answers
149
views
Product of elementary divisors
Let $A$ be an $(m \times n)$ integer matrix (if it helps, we can assume that a is a square matrix). Let $d_i,\ldots,d_s$ be the elementary divisors of $A$. I am interested in the product $\prod_{i=1}^...
- 1,101
1
vote
0
answers
82
views
Log convexity for the norm of a vector-valued function
Log convexity of various functions defined on the space of Hermitian matrices plays an important role in matrix analysis and probability theory.
Given $v \in \mathbb{C}^n$, $D$ a diagonal matrix with ...
- 31
1
vote
0
answers
77
views
notation for vector product in the space
The notation for vector (a.k.a. cross) product in $\mathbb{R}^3$ I usually see is $\times$.
However, some places use $\wedge$ instead, which IMHO creates a lot of confusion, as $\wedge$ usually is ...
- 14k
1
vote
0
answers
114
views
A simple Lie algebra with modules of a particular type
I’m trying to copy the following construction of P. Forster in group theory for Lie algebras. He takes a non-abelian simple group E which has an FpE-module V such that R = Rad(V ) is faithful and ...
- 310
1
vote
0
answers
639
views
On triangular Toeplitz matrices
Let $R(x)$ be the upper triangular Toeplitz matrix with first row $x$, so that $R_{ik}=x_{k-i+1}$ if $i\le k$ and $R_{ik}=0$ otherwise. Let $N(n)$ be the smallest number $N$ such that there exist $u_j,...
- 3,873
1
vote
0
answers
110
views
Extremal roots of Bernstein-Sato polynomials
Let $f \in \mathbb{C}[x_1,\ldots,x_n]$.
Consider $D[s]$, where $D$ is the ring of polynomial coefficient differential operators in $n$ variables, and $s$ is an additional formal variable.
Suppose $P(...
- 11
1
vote
0
answers
163
views
An identity satisfied by "Differentiation"
I asked this question in MSE but I did not received any answer. So I repeat it here:
Assume that $C$ is a coalgebra with comultiplication $\Delta:C \to C\otimes C$. The higher order ...
- 356
1
vote
0
answers
455
views
Possibility of Disconnected Subgraphs of a $k$ Connected $r$ regular Graph under a given condition
Context: Given a adjacency matrix A of a $r$-regular graph $G$ (not complete graph $K_{r+1}$) . $G$ is $k$ connected.
The matrix A can be divided into 4 sub-matrices based on adjacency of vertex $x ...
- 267
1
vote
0
answers
204
views
Complexity of reordering a matrix which consists independent sub matrices
Introduction:
Given a matrix A of a $k$ regular graph G. The matrix A can be divided into 4 sub matrices based on adjacency of vertex $x \in G$.
$A_x$ is the symmetric matrix of the graph $(G-x)$, ...
- 267
1
vote
0
answers
138
views
Minimum rank of certain matrices
Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in $\...
- 13.9k