0
votes
0answers
24 views

Smith normal forms and a math program

I am interested to know the Smith normal form of $4 \times 2$ matrices $M$: The two cases of my interests are: (1). $$M_1= \begin{pmatrix} 3 & 0\\ -5 & 4\\ 4 & -5\\ 0 & 3 ...
2
votes
2answers
76 views

How to find a basis for a tricky 2x 2 matrices vector space

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this ...
0
votes
1answer
38 views

Maximal Ideals of Matrix Ring

Let $D$ be a division algebra. I'm trying to show that all simple $M_n(D)$ modules are isomorphic to $D^n$. I know that $M_n(D)=D^n\oplus D_n \oplus \dots \oplus D^n$ (n terms). I have that if $S$ is ...
0
votes
0answers
18 views

Change of basis - free submodule

Let $R$ be a commutative ring and $(e_1, \dotsc, e_n)$ be a basis for $R^n$, the free $R$-module of rank n. Let $A$ be an $n \times n$ matrix with entries in $R$. Let $f_i = \sum \limits_{j=1}^n ...
6
votes
1answer
70 views

Describe all the quotient groups of $\mathbb{Z} \oplus \mathbb{Z}$

I'm thinking about a problem in algebraic topology of how to determine all the $k$-fold covers of $\mathbb{T}^2$, where $k$ is a certain integer. In thinking about the problem, I came upon the ...
0
votes
0answers
44 views

Simple module over matrix rings

I'm trying to prove that if $M$ is a simple module over $M_n(D)$ where $D$ is a division algebra, then $M\cong D^n$. I know that if $M$ is a simple module over $R$ then it is isomorphic to ...
0
votes
0answers
29 views

Identifying the abelian group with a presentation matrix

I am doing problems from Artin: \begin{bmatrix} 2 \\ 1\\ \end{bmatrix} and \begin{bmatrix} 2 & 4\\ 1 & 4\\ \end{bmatrix} For the First one after manipulating rows I ...
2
votes
1answer
50 views

Let $k$ be a division ring, then the ring of upper triangular matrixes over $k$ is hereditary

I'm reading Ring Theory by Louis H. Rowen, and he claimed that The ring of upper triangular matrices over a division ring is hereditary (it's on page 196, Example 2.8.13 of the book). I think it ...
4
votes
0answers
53 views

Symmetric non-degenerate bilinear forms over $\mathbb{Z}$ and $\mathbb{Q}$

Consider the four non-degenerate symmetric bilinear forms over $\mathbb{Q}$ given be the matrices $\bigl(\begin{smallmatrix} 1&0\\ 0&1 \end{smallmatrix} \bigr)$,$\bigl(\begin{smallmatrix} ...
2
votes
1answer
95 views

for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple

Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring ...
0
votes
1answer
44 views

Module, End(M), acts

What it means for a module $M$ and the endomorphism ring $\text{End}(M)$ that $\text{End}(M)$ acts diagonally on $M^n$, where $n$ is a positive integer? Also, if $G=(r_{ij})$ is a matrix with ...
2
votes
1answer
75 views

Can we classify commuting pairs of matrices up to conjugacy?

Recall that two $n\times n$ matrices over $\mathbb{C}$ are conjugate if and only if they have the same Jordan canonical form. Question. Is there a similar classification for commuting pairs of ...
0
votes
1answer
57 views

Free modules and free submodules. [duplicate]

Just finished my semester in advanced abstract algebra and there was one question that I could not answer. Let $R$ be commutative and let $(e_1,...,e_n)$ be a base for $R^{(n)}$. Put ...
4
votes
3answers
163 views

similar matrices over $\mathbb{Z}$

Let $A,B$ be $2\times 2$ matrices over $\mathbb{Z}$. Suppose $x^2+x+1$ is the characteristic polynomial for both $A,B$. Determine whether $A,B$ are similar to each other over $\mathbb{Z}$. How to ...
0
votes
1answer
137 views

matrices with same minimal and characteristic polynomials are conjugate

Let $A$, $B$ be two $n\times n$ matrices over $\mathbb{C}$. If $A$, $B$ have the same minimal polynomials and characteristic polynomials, then can we prove that $A$, $B$ are conjugate? i.e., does ...
6
votes
1answer
220 views

Obtain a basis of invertible matrices for $M_n(D)$, where $D$ is an integral domain

Let $D$ be an integral domain. Prove that $M_n(D)$ has a basis consisting of $n^2$ invertible matrices. Consider $M_n(D)$ as a $D$-module. Define invertible elements $V_{nn}=I_n$, $V_{n-1,n-1} = ...
3
votes
1answer
124 views

Similar matrices

Let $A$ be a complex $n×n$ matrix with $tr(A) = 0$. Show that there exists an invertible $n×n$ matrix $B$ such that all diagonal entries of $BAB^{-1}$ are zeros. $\bf Edit:$ Is the fact that $\forall ...
2
votes
2answers
279 views

Exercise in Jacobson's $Basic\ Algebra\ I$, Chapter 3

Well, I even don't understand the problem. Let $R$ be a ring and let $(e_1,\ldots,e_n)$ be a base for $R^{(n)}$. If I define: $$ f_j=\sum_{j'=1}^n a_{jj'}e_{j'} $$ for all $j \in ...
5
votes
2answers
415 views

How does one obtain the Jordan normal form of a matrix $A$ by studying $XI-A$?

In our lecture notes, there's the following example problem. Find a Jordan normal form matrix that is similar to the following. $$A=\begin{bmatrix}2 & 0 & 0 & 0\\-1 & 1 & 0 ...
3
votes
1answer
84 views

Does this complex remain exact after I restrict the maps?

$R$ is a commutative ring with unity. Assume you have two matrices $A:R^n\rightarrow R^m$ and $B:R^m\rightarrow R^n$ such that they form an exact complex in the obvious way, i.e., $$\cdots\rightarrow ...
3
votes
1answer
72 views

Hamiltonian Quaternions: A Call for Counterexample for ($AB=I_m \implies n≥m$)

How can I build a counterexample to $AB=I_m \implies n≥m$ in the ring of Hamiltonian quaternions? Notice that the vectors $(1,i)$ and $(j,k)=(1,i)j$ are linearly dependent as vectors of the right ...
3
votes
1answer
103 views

Question about Noncommutative Rings by I. N. Herstein

Here's an example that I do not quite understand it fully, it's on page 6 of the book. And here's what it says: Let $F$ be a field, and $F_n$ be a ring of all $n\times n$ matrices over $F$. We ...
1
vote
2answers
366 views

Finding invariant factors of finitely generated Abelian group

There is this question that I wasn't sure how to do but somehow got the answers partially correct (maybe). Suppose that the abelian group $M$ is generated by three elements $x,y,z$ subject to the ...
1
vote
1answer
71 views

Some questions on invariant factor decomposition of modules

Let $M$ be the $\mathbb{Z}$-module $F/N$ where $F=\mathbb{Z}^2$ and $N=\langle(6,4),(4,8),(4,0)\rangle\le\mathbb{Z}^2$. Consider the homomorphism $\varphi:\mathbb{Z}^3\to\mathbb{Z}^2$ such that the ...
1
vote
1answer
58 views

Some questions on Proof of Structure Theorem

I understand the general idea of the proof of Structure Theorem for finitely generated modules over a principal ideal domain but I found it quite difficult to follow some lines of reasoning in the ...
1
vote
1answer
782 views

How to programmatically find the inverse of a 2x2 matrix (mod 26).

I'm trying to create a hill cipher utility. One feature I want is to be able to compute the key if you have the plaintext and ciphertext. $C$ = ciphertext matrix ($2\times 2$), $P$ = plaintext matrix ...
4
votes
2answers
300 views

Rewriting automorphism of matrix algebra in terms of automorphisms of the underlying ring?

I've used the following idea as a black box for some time now, but it occurred to me I don't fully understand why it's true. Suppose $A=M_n(R)$ is the algebra of square matrices over some division ...
1
vote
0answers
101 views

When do the invariant factors of a direct sum of diagonal matrices correspond to those of its summands?

I am trying to prove something about matroids, which I have reduced to the following question: Suppose I have a matrix $M$ which is a direct sum of submatrices $M_1,M_2,\ldots,M_k$. When do the ...
4
votes
2answers
683 views

Classification of simple modules for algebra of upper triangular matrices?

I've been refreshing my linear algebra, and this is a question of curiosity I have. Let $U:=U_n(F)$ be the algebra of upper triangular $n\times n$ matrices over a field $F$. Is there a classification ...
1
vote
1answer
134 views

Characterizing matrices over a division ring as a product of a simple module?

I was reading about semisimple modules, and there's a fact relating to simple modules that I don't recall. Suppose you have a ring $R=M_n(D)$, by which I mean the $n\times n$ ring of matrices with ...
5
votes
2answers
342 views

On rank of a matrix whose entries are polynomials

(I took courses on linear algebra, but I don't know anything about $R$-modules or such things.) How do you define the rank of a matrix whose entries are polynomials in $K[X]$? If you assign some ...
2
votes
1answer
116 views

Checking Ruan axioms

I need to check only one axiom for matrix bimodule made from certain bimodule. Here some preparatory definitions. Matricial approach. (See here for details) For arbitrary linear space $V$ denote by ...
4
votes
1answer
98 views

Question about how linear independence of module elements gives conditions on the determinant of a matrix

I have been trying to prove linear independence of rows implies linear independence of columns in kind of an abstract setting of modules following some exercises and notes from an old course. The ...