All Questions
129 questions
26
votes
2
answers
3k
views
Singular values of sequence of growing matrices
I asked this question on math.stackexchange and haven't received an answer in two weeks, so I'm repeating it here.
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \cr
1/2 & 0 &...
- 656
1
vote
0
answers
221
views
Centralizer in a matrix algebra over commutative polynomials
Let $A=M_n(F[x_{ij}\mid 1\leq i,j\leq n])$ be the matrix algebra over commutative
polynomials in $n^2$ variables $x_{11},\dots,x_{nn}$ over a (nice enough) field $F$.
I would like to know what is the ...
- 179
4
votes
1
answer
233
views
Generating of the matrix ring by two hermitian matices
Let $p$ be a prime and $q=p^n$. Let $\mathbb F_{q^2}$ be a field with $q^2$ elements and $\sigma$ its authomorhism of order two. A $m$ by $m$ matrix $A$ over $\mathbb F_{q^2}$ is hermitian if $A^\...
0
votes
1
answer
189
views
trace cotrace Matrix
Hello I want to know whats mean (trace) cotrace matrix.
In the context, mapping a matrix (t x n) $\in$ GF($2^m$) to a cotrace matrix (tm x n) $\in$ GF($2$)?
- 1
2
votes
2
answers
242
views
Integer square $2 \times 2$ block matrix inverse
Let $\mathbf{M}$ be an integer square $2 \times 2$ block matrix
$$
\mathbf{M} =
\left(
\begin{array}{cc}
\mathbf{A} & \mathbf{B} \\
\mathbf{C} & \mathbf{D}
\end{array}
\right) ,
$$
where $\...
- 61
3
votes
1
answer
473
views
Relaxing commutativity. For c1,c2 find q1,q2: (1) [c1,c2]=q1c2-q2c1 (2) [q1,q2]=0, (3)...
Consider some elements c1,c2 in some ring.
Let me say that they are "relaxed commutative" if there exists two elements q1,q2,
such that the following conditions hold:
(1) $ [c_1,c_2]=c_1q_2-c_2q_1$
...
- 24.8k
5
votes
1
answer
1k
views
Algebra - Decomposition of a matrix polynomial
Dear All,
This is related with a problem that I'm trying to solve on my PhD dissertation in econometrics, and I thought that some mathmatician can know the answer.
What is known about a possible ...
- 51
16
votes
2
answers
4k
views
Inverse of a matrix over a non-commutative ring
What's the best algorithm to invert a matrix of non-commutative elements? In my case I have a matrix of matrices.
From first principals by equating the elements of M * M' to I (where M' is the inverse)...
- 361
51
votes
1
answer
2k
views
Invertible matrices over noncommutative rings
Let $A\in M_m(R)$ be an invertible square matrix over a noncommutative ring $R$. Is the transpose matrix $A^t$ also invertible? If it isn't, are there any easy counterexamples?
The question popped up ...
- 2,941
23
votes
4
answers
2k
views
A matrix algebra has no deformations?
I have often heard the slogan that "a matrix algebra has no deformations," and I am trying to understand precisely what that means. While I would be happy with more general statements about finite-...
- 8,559
8
votes
4
answers
1k
views
Doubly stochastic matrices as squares of entires of unitary matrices
Given a unitary matrix $A$ with entries $a_{ii}$, it's clear that the matrix $B$ with entries $b_{ii} = |a_{ii}|^2$ is doubly stochastic. Is the inverse of this statement true? Namely, given a ...
- 125
10
votes
2
answers
1k
views
When the determinant of a 2x2 polynomial matrix is a square?
Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...
- 1,488
9
votes
1
answer
709
views
Automorphisms of a matrix in Smith normal form?
Added: Amritanshu Prasad's answer makes it clear that I am really asking for a description of the group of integer unimodular matrices $P$ such that $D^{-1}PD$ is also integer. These matrices are ...
- 2,150
9
votes
3
answers
670
views
Algebraic axiomatization for AB+BA^T operation on matrices
Let us consider a matrix algebra $\operatorname{Mat}_{n\times n}(K)$, where $K$ is a field, $\operatorname{char} K \neq 2$.
It is well-known that the axiomatization of commutator operation $[A,B]=AB-...
- 413
9
votes
3
answers
2k
views
On similar matrices and polynomial matrices
I'm teaching linear algebra and I'm encountering this theorem:
two matrices $A$ and B are similar iff $tI - A$ and $tI - B$ are equivalent (as polynomial matrices), where $I$ is the unit matrix.
The ...
- 357
3
votes
1
answer
303
views
ABA-product of matrices and length of chains of principal inner ideals
Let $k$ be a field, $p,q$ positive integers, and let $R$ be the space of $(p \times q)$-matrices over $k$, and $S$ be the space of $(q \times p)$-matrices over $k$. For every matrix $A \in R$, we ...
- 6,614
5
votes
2
answers
1k
views
Canonical form for a pair of quadratic forms
Could anyone recommend a reference to a canonical form for a pair of quadratic forms over R (not necessarily positively definite)? This is probably related to the Weierstrass elementary divisors (but ...
- 1,488
3
votes
1
answer
218
views
decompositions of matrices over $\mathbb{Q}$
Given a matrix $A\in GL_n(\mathbb{Q})$. Can it be expressed as a product of two matrices $B,C$ with $B\in GL_n(\mathbb{Z}[1/p])$ and $C\in GL_{n}(\mathbb{Z}_{(p)})$, where $ \mathbb{Z_p}$ denotes the ...
- 11.1k
4
votes
2
answers
483
views
Rank of sum of Galois conjugates of a matrix
Given an invertible square matrix $M$ with entries from some number field $K$ which is Galois over $\mathbb{Q}$, sum the Galois conjugates of $M$ to form a new matrix $M' = \Sigma_{\sigma \in \mathrm{...
- 1,810
22
votes
1
answer
33k
views
vector to diagonal matrix [closed]
For any column vector we can easily create a corresponding diagonal matrix, whose elements along the diagonal are the elements of the column vector.
Is there a simple way to write this transformation ...
- 247
2
votes
1
answer
3k
views
Is it possible to decompose a symmetric, positive definite matrix in this way?
Let $\Sigma$ be a symmetric positive definite matrix. Then the Cholesky decomposition gives us $\Sigma=LL'$ where $L$ is lower triangular and unique.
Under what conditions (if any) does there exist ...
- 269
2
votes
4
answers
2k
views
Semi-simple matrices over fields of finite characteristic
Well-known and useful facts are:
any symmetric matrix over $\mathbb R$ is semi-simple (i.e. diagonalizable), and
any hermitean matrix over $\mathbb C$ is semi-simple.
I will loosely speak about the ...
- 25.5k
18
votes
1
answer
1k
views
Analogue of Smith normal form for matrices over $\mathbb Z[t]$
Let $R$ be a principal ideal domain and $A \in M_n R$. It is well known that there exist invertible matrices $Q$ and $S$ and a diagonal matrix $D= {\rm diag}(a_1,\dots,a_n)$ such that
$a_i \mid a_{i+...
- 25.5k
0
votes
0
answers
177
views
Exotic isomorphism of matrix rings (2)
Let $R,S,T$ be (associative, with a 1) rings, and $m,n,r,s$ be non-negative integers.
If
(a) $R$ is isomorphic to $M_{m\times m}(T)$ and $S$ is isomorphic to $M_{n\times n}(T)$ and $m\cdot r = n\...
user5810
6
votes
3
answers
3k
views
Defining Multiplication in Polynomials over Rings of Matrices
More explicitly, if $M_{2 \times 2}(\mathbb{R})[x]$ denotes the ring of polynomials over the ring of 2x2 matrices with real coefficients (with indeterminate x a 2 by 2 matrix with real coefficients), ...
- 370
2
votes
2
answers
492
views
on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring
Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some $...
- 1,422
17
votes
1
answer
710
views
Standard polynomials applied to matrices
The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by
$${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\...
- 52.3k
28
votes
6
answers
5k
views
Expressing $-\operatorname{adj}(A)$ as a polynomial in $A$?
Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$.
I ...
- 427
2
votes
1
answer
445
views
Algebra of endomorphisms of f.g. modules as subquotients of matrix algebras
Let $A$ be a $C$-algebra, where $C$ is a commutative ring with $1$, and $M$ be a finitely generated left $A$-module.
Question: Is it true that we can always find a positive integer $n$, a $C$-...
- 279