# Tagged Questions

**11**

votes

**1**answer

172 views

### Hlawka inequality for determinants of positive definite matrices

It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) ...

**1**

vote

**1**answer

98 views

### How to calculate $(A^{-1})_{ii}$ for an invertible hyperhermitian quaternionic matrix $A$?

The article
Alesker, S. (2003). Quaternionic Monge-Ampere equations. The Journal of Geometric Analysis, 13(2), 205-238.
has the following CLAIM:
Claim. Let $A$ be an invertible hyperhermitian ...

**-2**

votes

**1**answer

317 views

### For a ring $A$, is $A$ Morita equivalent to $M_\infty(A)$? [closed]

Let $A$ be a ring, let $M_n(A)$ be the ring of $n$-by-$n$ matrices with elements in $A$, $A$ is Morita equivalent to $M_n(A)$, I was wondering if this also applied to infinite matrices? That is, if ...

**4**

votes

**1**answer

294 views

### Does $\text{Mat}_n(k)$ have some universal properties similar to its universal enveloping algebra?

Let $k$ be a field and $\text{Mat}_n(k)$ be $n \times n$ matrices over $k$. Let's consider $\text{Mat}_n(k)$ as an associative algebra and denote $gl_n(k)$ be the same $k$-linear space as ...

**0**

votes

**1**answer

116 views

### Number of minimal left ideals

Is there any way to compute the number of minimal left ideals of $M_n(K)$, the full $n\times n$ matrix ring with entries in the field $K$ ?

**3**

votes

**1**answer

195 views

### Polynomial identities for mod p matrices

Can there be a polynomial over the field $F_p$ of $p$ elements ($p$ prime) in non-commuting variables $X_1,..., X_r$ such that:
1) $f(A_1,...,A_r)=0$ for every $n \times n$ matrices $A_1,...,A_r$ ...

**15**

votes

**1**answer

385 views

### Geometry of numbers for three by three matrices?

While trying to use Minkowski's theorem to calculate the (left) class number of a noncommutative ring, I ran into the following problem:
What is the volume of the largest symmetric convex subset ...

**3**

votes

**1**answer

192 views

### Existence of a generalized matrix inverse over an arbitrary field?

Let $A\in M_n(K)$ be a square matrix over a field $K$. The notion of inverse matrix was generalized by Moore and Penrose for real and complex matrices
(also called pseudo-inverse $A^{\dagger}$ of $A$, ...

**2**

votes

**0**answers

116 views

### when upper triangular matrix modulo prime ideals implies upper triangular?

Let $E/ \mathbb{Q}_p$ be a finite extension, let $\mathcal{O}$ be the ring of integers of $E$. Let $A$ be a reduced noetherian local complete $\mathcal{O}$-algebra with the maximal ideal ...

**9**

votes

**0**answers

469 views

### Is “being a full ring of quotients” a Morita invariant property?

Definition and context:
An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...

**12**

votes

**1**answer

636 views

### Number of idempotent $n\times n$ matrices over $\mathbb{Z}_m$ ?

Is there any known formula for the number of idempotent $n\times n$ matrices over $\mathbb{Z}_m$ ?
The number of idempotent matrices over a finite field is well-known and since we can decompose $m$ ...

**5**

votes

**4**answers

1k views

### Proving a determinant = 0

The two most elementary ways to prove an N x N matrix's determinant = 0 are:
A) Find a row or column that equals the 0 vector.
B) Find a linear combination of rows or columns that equals the 0 ...

**0**

votes

**0**answers

132 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 ...

**4**

votes

**1**answer

198 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 ...

**0**

votes

**1**answer

143 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($2$)?

**3**

votes

**1**answer

354 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$
...

**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 ...

**6**

votes

**2**answers

1k 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 ...

**34**

votes

**1**answer

866 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 ...

**17**

votes

**4**answers

1k 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 ...

**6**

votes

**4**answers

675 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 ...

**9**

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 $A=A_1 A_2 â€¦ A_{2n}$, in ...

**8**

votes

**1**answer

545 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 ...

**9**

votes

**3**answers

601 views

### Algebraic axiomatization for AB+BA^T operation on matrices

Let us consider a matrix algebra $Mat_{n\times n}(K)$, where $K$ is a field, $char K \neq 2.$
It is well-known that the axiomatization of commutator operation $[A,B]=AB-BA$ on matrix algebra leads us ...

**6**

votes

**2**answers

2k views

### on similar matrices and polynomial matrices

I m teaching linear algebra and encounter 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 proof that I ...

**3**

votes

**1**answer

190 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 ...

**4**

votes

**2**answers

829 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 ...

**3**

votes

**1**answer

199 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 ...

**3**

votes

**2**answers

328 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 ...

**2**

votes

**1**answer

648 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 ...

**2**

votes

**4**answers

732 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 ...

**15**

votes

**1**answer

775 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 ...

**0**

votes

**0**answers

152 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 = ...

**3**

votes

**3**answers

851 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), ...

**2**

votes

**2**answers

375 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 ...

**15**

votes

**1**answer

568 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 ...

**1**

vote

**1**answer

199 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 ...