0
votes
1answer
41 views

Tensor product of Frobenius algebras

In proving the fact that the tensor product of any two finite-dimensional Frobenius algebras $R$ and $S$ over the same field $k$, it is usually defined a $k$-bilinear pairing $E: W×W→k$ where ...
4
votes
1answer
51 views

Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $

I'm having a hard time counting. I need to count the number of elements for the multiplicative group of invertible $2\times 2$ matrices $GL_{2}\left( \mathbb{F}_{p}\right) $ with elements from the ...
17
votes
2answers
179 views

Show that the kernel of the map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion.

I am trying to show that the kernel of the natural map $SL(n, \mathbb{Z}) \to SL(n, \mathbb{Z}/3\mathbb{Z})$ has no torsion. That is, if $A$ is in the kernel then $A = I$ or $A^n \neq I$ for all $n ...
0
votes
0answers
13 views

Show that U subspace is supplementary to the kernel. How to find values of a b c d using intersection of two matrices.

I already found the kernel to be \begin{pmatrix} -2c&-2d\\c&d \end{pmatrix}. and U is a subspace of a $M_2$ matrix defined by \begin{pmatrix} a&b\\2a&2b \end{pmatrix}. So i have to ...
1
vote
1answer
41 views

Jordan-Chevalley decomposition of $T$ acting on $k[T]/(\pi(T)^e)$

Given an algebraically closed field $K$, a f.d. vector space $V$ over $K$ and $A\in{\rm GL}(V)$, we can view the space $V$ as a $K[T]$-module, where $T$ acts by $A$. Using the fundamental theorem of ...
1
vote
2answers
32 views

How to find a basis of an image of a linear transformation?

I apologize for asking a question though there are pretty much questions on math.stackexchange with the same title, but the answers on them are still not clear for me. I have this linear operator: ...
2
votes
2answers
51 views

Can units in $M_n(\mathbb{Z})$ be moved to the other side?

Let $M, U_1 \in M_n(\mathbb{Z})$ with $U_1$ a unit (i.e. $\lvert \det(U_1) \rvert=1$). Can I always find another unit $U_2\in M_n(\mathbb{Z})$ such that $U_1 M = M U_2$?
1
vote
1answer
41 views

Showing a matrix is nilpotent if its charateristic polynomial is $t^n$ mod ${\rm nil}(R)$

Let $R$ be a commutative ring. How to prove the following: If $\chi_A(t) \equiv t^n \bmod\operatorname{nil}(R)$ then $A \in M_n(R)$ is nilpotent. Note $\chi_A$ is the characteristic polynomial ...
2
votes
0answers
21 views

Find a reduced echelon basis from a reduced echelon matrix.

The reduced row matrix was this ---> $\begin{pmatrix}1&2&0&1&0\\0&0&1&3&0\\0&0&0&0&1\\0&0&0&0&0&\end{pmatrix} = 0$ So i computed ...
0
votes
1answer
19 views

Prove that the direct sum of a symmetric and skew symmetric matrix belongs to $M_n(K)$ using $A_{ij}$ and $A_{ji}$ notation.

Basically Let $M_n(K)$ be an $n\times n$ matrix of a $K$ vector space. $U =\{A\in M_n(K)\;|\;A_{ij}=A_{ji}\}$ $W =\{A\in M_n(K)\;|\;A_{ij}=−A_{ji}\}$ So I don't understand my mark scheme. It says ...
1
vote
2answers
43 views

Show that 2 matrices belong to a square matrix by taking the transpose. Vector spaces

Let $M_n(K)$ be an $n\times n$ matrix of a K vector space. \begin{align} U &= \{A ∈ M_n(K) | A_{ij} = A_{ji} \} \\ W &= \{A ∈ M_n(K) | A_{ij} = -A_{ji} \} \end{align} Prove that $U$ and $W$ ...
2
votes
1answer
84 views

Projective special linear groups are co-hopfian?

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
3
votes
2answers
69 views

Is there a way to determine the matrix of $\Lambda^k(T)$ given the matrix of $T$?

Let $T$ be an endomorphism of a finite dimensional vector space $V$. Suppose that $(v_1,\ldots v_n)$ is an ordered basis of $V$. And let $[T]$ be the matrix of $T$ with respect to this basis. Is ...
0
votes
0answers
45 views

A problem on matrices over a commutative ring

Let $M_{m,n}(R)$ denote an $m\times n$ matrix with each entry over a commutative ring $R$, $m\leq 2\leq n$, and there is a matrix $\mathbf{B} = M_{m,n}(R)$. $\mathbf{B}\mathbf{s} = \mathbf{a}$, where ...
0
votes
0answers
99 views

Is there some fast and efficient way for solving $x$

Let $b$ be a given constant scalar between 0 and 1, and $A$ a given $N \times N$ transitional probability matrix (i.e., each row sum of $A$ is 1, and $0\le {A}_{(i,j)} \le 1$). Let $A\circ A$ denote ...
6
votes
1answer
61 views

Matrix restoring (modulo n)

Let $m,n\geqslant 2$, and $A\in \mathcal{M}_n(\mathbb Z)$ such that $\det A \equiv 1 \pmod m$. Does it (necessarily) exist $M\in \mathrm{GL}_n(\mathbb Z)$ such that $A\equiv M \pmod m$?
1
vote
1answer
42 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
0
votes
1answer
37 views

Show that $\lambda_1 = \min \{ Q(u) \mid \|u\| = 1 \}$ and $\lambda_m = \max \{ Q(u) \mid \|u\| = 1 \}$

Let $V$ a vector space over $K$ and $Q(u) = \langle u, Tu \rangle$ a quadratic form. $T$ is a symmetric operator. The eigenvalues of $T$ are sorted by size $\lambda_1 < \dots < \lambda_m$. How ...
1
vote
2answers
56 views

Write Matrix $A$ to $A = \sum_{i=1}^{3} \lambda_i P_i$

Let $A$ be a Matrix: $$A = \begin{pmatrix} 1 & 0 & 3i \\ 0 & -3 & 0 \\ -3i & 0 & 1 \end{pmatrix}$$ Now I want to write $A$ as $$A = \sum_{i=1}^{3} \lambda_i P_i$$ I ...
2
votes
2answers
53 views

How to bring $5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$ to the form $\langle x',Ax' \rangle = 1$

How can I bring $$5x_1^2 - 26x_1x_2 + 5x_2^2 + 10x_1 - 26x_2 = 31$$ to the form $$\langle x',Ax' \rangle = 1$$ where $x' = \alpha x + \beta$ where $\alpha \in \mathbb{R}^+$ and $\beta \in ...
1
vote
2answers
62 views

Show that $\langle x, Ax \rangle + \langle b, x \rangle = c$ can be transformed to $\langle x', Ax' \rangle = 1$

Let $A$ be a real, regular, symmetric $n \times n$ matrix, $b \in \mathbb{R}^n$ and $c \in \mathbb{R}$ How can I show that $$\langle x, Ax \rangle + \langle b, x \rangle = c$$ can be transformed by ...
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 ...
0
votes
0answers
37 views

Canonical Forms For Matrices

In the following paper by Wedderburn what are the restrictions on the field $\mathbb F$ or on the linear application $\varphi$ that the author refers to obtain the matrix B? ...
6
votes
1answer
100 views

Existence of $p \times p $ matrices $A$ and $B$ over the field $\mathbb F_p$, $p$ prime, such that $AB-BA=I$. [duplicate]

Let $p$ be a prime number. Prove or disprove that there exists $p\times p$ matrices $A$ and $B$ over a field $\mathbb F_p$ with $AB-BA = I$. With the aid of MAPLE i was able to find out that ...
1
vote
1answer
40 views

Linear transformation matrix representation with differentiation answer confirmation

I hope you liked the title. I have a question that is as follows: Consider the linear transformation $T: P_3(\mathbb{R}) \to P_3(\mathbb{R})$ given by $$T(f(x))=f(0)+f'(x)+f''(x)$$ Where the ...
4
votes
1answer
84 views

Finding a basis for $\ker(T)$

I have this question: Let $Z\in M_{2\times2}(\mathbb{R})$ be defined as $$Z = \left( \begin{align} 1 &&1\\1 &&1 \end{align} \right)$$ and consider $T: ...
3
votes
2answers
37 views

Basis and dim of the set of all $n\times n$ symmetric matrices.

An $n \times n$ square matrix $A$ is called symmetric if $A^T = A$ Show that the set of all $n \times n$ symmetric matrices, denoted $S$, is a subspace of $M_n(\mathbb{R})$. Give a basis for $S$ ...
2
votes
1answer
156 views

Principal ideal ring

Let $K$ be a principal ideal ring. How to prove that for any $ x= (x_1, x_2)^t \in K^2 $ there exists a matrix $G \in SL_2(K)$ such that $Gx = (\gcd(x_1, x_2),0)^t $ ?
1
vote
0answers
37 views

Complex matrices: looking for homomorphism

Let $\mathbb{C}$ denote the complex numbers, and let $M_2(\mathbb{R})$ be the ring of $2$ by $2$ matrices with real entries. Define a function $f:\mathbb{C} \to M_2(\mathbb{R})$ by $ f(a+bi) = ...
1
vote
2answers
58 views

Conjugates of the upper triangular matices

It's a shame...I want to give an explicit description of the set $\bigcap_{m \in GL_n(K)} mUm^{-1}$, $U$ being the upper triangular subgroup of $GL_n(K)$. It seems to be $K^\times I_n$ but I do not ...
1
vote
0answers
41 views

$(\sigma_0^{-1} SL(2,\mathbb{Z}) \sigma) \cap SL(2, \mathbb{Z})$ is a right coset of $\Gamma_0(m)$ in $SL(2, \mathbb{Z})$

Let $\Gamma_0(m)$ be the subgroup of $SL(2,\mathbb{Z})$ which is defined as follows: $$\Gamma_0(m)=\left\{ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in SL(2, \mathbb{Z} : c ...
2
votes
0answers
82 views

Tensor product of the spaces of quaternions and complex numbers

Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b ...
2
votes
1answer
38 views

how can I obtain the lower bound for this case?

For an arbitrary pair of orthogonal bases $\phi$, $\varphi$ that construct the matrix A and which of them are $n \times n$. mutual coherence for matrix $A$ is defined as te maximal inner product ...
2
votes
0answers
67 views

Application of Schur's Lemma

Given three matrices , $B_1 \in Mat_{n_1}(k)$, $B_2 \in Mat_{n_2}(k)$, and $C \in Mat_{n_1 \times n_2}(k)$, where $k$ is a perfect field, and $B_1 $ and $B_2$ are irreducible matrices. Required to ...
1
vote
2answers
38 views

When is a companion matrix diagonalizable and what does this say about the associated field extension?

Consider the $n\times n$ matrix $$ M=\begin{pmatrix} 0 & 0 & 0 & \cdots & 0 & 0 & -c_0\\ 1 & 0 & 0 & \cdots & 0 & 0 & -c_1\\ 0 & 1 & 0 & ...
3
votes
0answers
81 views

How can we solve this question without brute force

If $A\in GL_n(R)$, where $R$ is a commutative ring with identity, I would like to prove $$ M=\begin{pmatrix} A & 0 \\ 0 & A^{-1} \\ \end{pmatrix}\in ...
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 ...
6
votes
1answer
419 views

How to prove that the inverse of a matrix is unique?

The ring of matrix is not an integral domain. How to prove that the inverse is unique?
1
vote
1answer
34 views

Is the operation associative

Is it known that the multiplication of matrices is a associative operation ? So,is the relation $(A \cdot B) \cdot C=A \cdot (B \cdot C)$ true?? ($A,B,C$ are matrices)
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 ...
2
votes
1answer
32 views

Matrix Groups in Abstract Algebra

QUESTION: Let $h= \begin{pmatrix} -1 & 1\\-1&0 \end{pmatrix} \in GL_2(\Bbb R)$. Find $\langle h\rangle$. I'm stuck on the solution, but here is what I have: Let $h=\begin{pmatrix} ...
8
votes
2answers
103 views

Order of invertible matrices

I recently came across an interesting problem in Artin which says: If $A \in GL_2(\mathbb{Z})$ is of finite order then it has order $1,2,3,4,6$. I was looking for a generalization of this problem. ...
1
vote
1answer
47 views

Decomposing the set of $2 \times 2$ complex matrices into orbits under left multiplication

I have some issues with a problem which is asking me to decompose the set of $2 \times 2$ complex matrices $\mathbb{C}^{2 \times 2}$ in orbits under the left multiplication operation on the group ...
1
vote
1answer
36 views

Linear Algebra, matrix representation

Let $V$ be a vector space of all real differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ with basis $(e^{3t}, te^{3t},t^2e^{3t})$. Let $D: V \rightarrow V$ be the derivation operator on ...
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
1answer
55 views

Prove that the group of $3\times3$ rational unipotent triangular matrices modulo its center is isomorphic to the additive group $\mathbb Q^2$

Let $G$ be the group of matrices of the form: $$\left( \begin{array}{ccc} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{array} \right)$$ with $a,b,c \in ...
0
votes
1answer
24 views

suggest me a example for non singular, conjugate-symmetric sesquilinear form ????

I only know that fact that the matrix corresponding to the non singular, conjugate-symmetric sesquilinear form is a unitary matrix. and SU_n(q) is the unitary groups is the collection of the ...
2
votes
1answer
94 views

Abstract Algebra Matrix Group Theory

The matrix group G = SL(n, $\mathbb{R}) = \{A \in M(n, \mathbb{R})\} \text{ acts on } X= R^n$ by left matrix multiplication: $\tau _A(x) = A\cdot x (\text{matrix product }(n \times n ) \cdot (n \times ...
-1
votes
1answer
27 views

Singular matrices over a commutative ring $R$, with a given adjoint matrix

First, I apologize if this is a duplicate question. I also must apologize if this has a trivial solution. This question has two parts: Let $R$ be a commutative ring with $1$, and let $F = R^n$ be a ...