All Questions
Tagged with ra.rings-and-algebras linear-algebra
252 questions
17
votes
3
answers
905
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 2,071
5
votes
3
answers
757
views
Unital *-homomorphisms between matrices
It is mentioned on Wikipedia that every unital *-homomorphism $\Phi:M_i\to M_j$ is necessarily of the form $\Phi(a)=U^*(a\otimes I_r)U$ for some unitary $U$ and some $r$. (Here $M_i$ are the $i\times ...
- 896
29
votes
3
answers
2k
views
Categorification of determinant
The notion of trace of a matrix can be generalized to trace of an endomorphism of a dualizable objects in a symmetric monoidal category. (See Ponto & Shulman for a nice description.)
Is there a ...
- 290
7
votes
2
answers
589
views
On a matrix problem in the field $\mathbb F_2$
Given $M$ a symmetric matrix in $\mathbb F_2^{n\times n}$ having $\mathsf{det}_\mathbb R(M)\neq0$ (non-singular in reals) and satisfying $PMP'=(M+J+I)$ or $P(M+J+I)P'=M$ where $P$ is a permutation ...
- 13.9k
5
votes
2
answers
395
views
Are there examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras?
Are there any examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras? Or can one show that none exist?
I went through this list of all complex associative ...
- 3,001
0
votes
0
answers
249
views
What is the computational complexity of solving a highly underdetermined system?
Let $F$ be a finite field with $q$ elements. Consider an underdetermined system of linear equations with $m$ equations and $n$ variables where $n\gg m$. What is the complexity of solving such a highly ...
- 131
1
vote
2
answers
349
views
How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there?
How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there?
As much as I have searched, I have not found any results that answer my question; not even for k = 1,2.
5
votes
0
answers
241
views
Growth rate of cohomology
Fix a finite dimensional graded vector space $V$, a differential $d$ on $V \otimes V $ $(i.e.\,\, d^2=0\,\, and \,\, deg\, d =1)$ such that $(d_{1,2} \otimes id_3)$ anti-commutes with $((-1)^{deg}_1 \...
- 189
8
votes
1
answer
341
views
General Sylvester's linear matrix equation
For what conditions on $A$, $B$ and $C$ (square matrices of size $n$) would there be a unique solution to
$$
ABX + AXC + XBC = D,
$$
for any $D$? Can one expect a characterization similar to the ...
- 1,216
3
votes
1
answer
418
views
Quaternions as eigenvalues of rank 3 tensors
Let us consider a matrix $M^{(a)}$ of size $N \times N$, having $N$ eigenvalues $\lambda_i \in \mathbb{C}$.
Considering a rank-3 tensor, we can informally think of it as a sequence of $N$ matrices $M^{...
- 117
5
votes
1
answer
416
views
Is every endomorphism a linear combination of projections?
Let $E$ be a vector space over a field $K$. If $u \in \mathscr L(E)$ is an endomorphism of $E$, can it be written as a linear combination of projections (i.e. endomorphisms $p$ of $E$ such that $p \...
- 615
0
votes
1
answer
215
views
Principal minors and similarity
Given two real and irreducible matrices $A$ and $B$ of size $n \times n$. A matrix $A$ is irreducible if there is no permutation matrix $Q$ so that
$$
Q^{-1} A Q = \begin{bmatrix} E & G \\ 0 & ...
- 909
9
votes
1
answer
472
views
$M = AA^t$ where $A$ has unit norm columns
Let $M \in \mathbb{R}^{k\times k}$ positive definite with $\operatorname{tr} M = m$, where $m$ is an integer such that $m \geq k$. I have found a way (using this answer) to decompose $M = AA^t$ with $...
- 185
6
votes
1
answer
328
views
Algebra with a certain abelian group as the multiplicative group
Let $A$ be an abelian group. Are there an algebra $\mathfrak{X}(A)$ s.t. the multiplication group is isomorphic to A ? i.e.
$$
\mathfrak{X}(A)^{\times} \simeq A.
$$
For example, for $A=\mathbb{Z}/4\...
- 479
2
votes
1
answer
220
views
Two equivalent matrices? [closed]
By coincidence I noticed that the following two matrices yield the same eigenvalues
\begin{pmatrix} A & B \\ B^* & A \end{pmatrix} and \begin{pmatrix} 0& A+b1_{\mathbb C^{2 \times 2}} \\ A+...
- 536
2
votes
1
answer
290
views
Endomorphism rings of infinitely generated free modules generated by idempotents?
Let $M$ be a free right $R$-module. When $M_R\cong R_R^n$ with $n\in \mathbb{Z}_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M_R)$ is isomorphic to $\mathbb{M}_n(R)$. We also know ...
- 23
10
votes
1
answer
807
views
How many Lie and associative algebras over a finite field are there?
This question is related to the following general question:
Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...
- 398
2
votes
3
answers
324
views
Efficient algorithm for matrix equation $AXB + BXA = F$
For $n\in\mathbb{N}$, let $A\in\mathbb{R}^{n\times n}$ and $B\in\mathbb{R}^{n\times n}$ be two symmetric positive definite matrices and let $F\in\mathbb{R}^{n\times n}$ be arbitrary.
Is there any ...
- 205
2
votes
0
answers
114
views
Projective group of Plucker quadric over the reals
A somewhat elementary question but seemingly difficult to find a suitable reference:
Consider the six-dimensional real space $\wedge^2(\mathbb R^4)$ with basis $e_i \wedge e_j \ (i < j)$ where $...
- 356
0
votes
0
answers
141
views
Stabilizers in the action of $\mathrm{GL}(n, \mathbb Z)$ on $\mathbb Z^n$
How can we calculate effectively the subgroups of $G: = \mathrm{GL}(n, \mathbb Z)$ which fix pointwise a given submodule $S$ of $\mathbb Z^n$ in the action of $G$ on $\mathbb Z^n$ by left ...
- 356
5
votes
0
answers
181
views
Is there a list of all real unital subalgebras of M(2,C)?
Is there a complete classification of all real unital subalgebras of $M(2,\mathbb C)$ up to isomorphism? The list should include $M(2,\mathbb C)$, the quaternions, complex numbers, split-complex ...
- 4,943
15
votes
4
answers
869
views
What is known about ordinary character values at involutions?
Let $G$ be a finite group and let $\chi$ be the character of an irreducible complex representation $\rho$ of $G$ on $V$.
Let $x$ be an involution in $G$.
I'd like to ask the following
Question 1:
...
- 1,755
3
votes
0
answers
39
views
A non-singularity property for sets of real matrices
Let $M_N(\mathbb{R})$ be the ring of $N\times N$ real matrices. We say that a couple $(\mathcal{U},\mathcal{V})$, with $\mathcal{U},\mathcal{V}\subseteq M_N(\mathbb{R})$ is admissible if, for every $A\...
- 943
16
votes
0
answers
574
views
Are $0, 1, 4, 7, 8$ the only dimensions in which a bivector-valued cross product exists?
It is a well-known mathematical curiosity that ordinary (vector-valued) cross products over $\mathbb{R}$ exist only in dimensions $0, 1, 3$ and $7$ (this fact is related to Hurwitz's theorem that real ...
- 1,206
44
votes
2
answers
2k
views
Fermat's Last Theorem for integer matrices
Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
- 1,889
6
votes
0
answers
192
views
Bar notation in Bourbaki’s *Lie groups*, Chap. IX
I am looking at Chapter IX (Compact Real Lie Groups), §4, Exercise 8 (translation). Given a complex subspace $\mathfrak p$ in the complexification $\mathfrak g_{\mathbf C}$ of some $\mathfrak g$, they ...
- 31.5k
8
votes
2
answers
808
views
Bounding the spectral gap of a simple symmetric matrix
I have a seemingly innocent linear algebra problem that I cannot solve, and which I hope that you would kindly offer some insight into. Here is the description: Let $\mathbf{a} = (a_1, a_2, \dots, a_d)...
- 333
2
votes
1
answer
120
views
Can Hom-Lie algebras be seen as an $\Omega$-algebras?
An $\Omega$-algebra over a field $K$ is a $K$-algebra $A$ with a set of multilinear operators $\Omega$, where $\Omega=\bigcup_{m=1}^{\infty} \Omega_{m}$ and each $\Omega_{m}$ is a set of $m$-array ...
- 367
5
votes
0
answers
91
views
elementary matrices over a regular ring
Let $n\in\mathbb{N}$ with $n\geq 3$. Let $A$ be a regular ring and $\gamma\in GL_{n}(A)$. We suppose that $\gamma$ is homotopic to the identity, i.e. there exists $\alpha\in GL_{n}(A[X])$ such that $\...
- 3,472
4
votes
2
answers
360
views
Double centralizer in special linear algebra
It is well known that for a matrix $A$ in $\mathfrak{sl}_n(\mathbb{C})$, we have the following equivalence:
$$\dim Z(A) \text{ is minimal} \leftrightarrow A \text{ is cyclic}$$
where $Z(A)$ is the ...
- 617
7
votes
3
answers
1k
views
Linear independence of element-wise powers of positive vectors
Consider a vector $x$ with $0 < x_1 < \cdots < x_n < \infty$, and let $0 < \gamma_1 < \cdots < \gamma_n < \infty$.
I would like to show that $x^{\gamma_1}, \ldots, x^{\...
- 73
3
votes
1
answer
454
views
Difference of pseudoinverse bound under assumptions
This seems like a standard problem, but unable to find a solution online.
Suppose we have two singular PSD matrices A and B with the following assumptions:
$ 0 < x \leq ||A|| \leq y$
$ 0 < ||...
- 33
4
votes
1
answer
153
views
Linear maps preserved by algebra automorphisms
Let $A$ be a finite dimensional , associative, unital $F$-algebra, where $F$ is a field.
Let $s_A:A\to F$ be an $F$-linear map.
Now consider an arbitrary field extension $K/F$, and define $s_{A\...
- 1,766
3
votes
0
answers
180
views
Automorphisms of infinite matrix algebra
This is a similar question to one that I posted in MSE a few days ago.
I recently came across this paper from Alahmedi, Alsulami, Jain and Zelmanov, which quoted the following result for $M_\infty(K)$...
- 195
5
votes
2
answers
706
views
Zero tensor product over a complex algebra?
Let $A$ be an algebra over $\mathbb{C}$. Let $M$ be a left $A$-module, let $N$ be a right $A$-module and consider the tensor product $N \otimes_A M$, which is a complex vector space.
Q1: Can this ...
- 9,853
2
votes
0
answers
61
views
Generalizing polynomial identities for rings
For a ring $R$, a polynomial identity of $R$ is a polynomial (in non-commuting variables) $f(x_1,\ldots,x_n)\in \mathbb{Z}[x_1,\ldots, x_n]$ such that for any choice of $a_i\in R$, $f(a_1,\ldots, a_n)=...
- 21
10
votes
0
answers
237
views
Generalized eigen property of a matrix
Given a $n \times n$ invertible matrix $A$, I am interested in the set
$$
\mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}.
$$
Thus, for all eigenvalues $\lambda_i$, we have $...
- 909
2
votes
1
answer
207
views
Question on the proof of any finite dimensional module of a semisimple Lie algebra is semisimple
I'm going through a proof of the theorem, any finite dimensional module of a semisimple Lie algebra is semisimple, from page 10 of these notes (pdf). I am having a difficult time understanding few ...
- 3,625
7
votes
0
answers
168
views
Matrix operations preserving the middle coefficients of characteristic polynomial
Let $n$ be a positive integer. For a ring $A$ and matrix $M \in \mathrm{Mat}_{n \times n}(A)$, let $\chi_{M}(t) = \det(M-t \operatorname{id}_{n}) = (-1)^{n}(t^{n} - \sigma_{M,1}t^{n-1} + \dotsb + (-1)^...
- 428
4
votes
0
answers
129
views
How to formulate supercommutativity in a characteristic free way?
I would never dare posting this here, but the question https://math.stackexchange.com/q/3019853/214353 on math.SE did not receive any feedback (except for 13 views and one upvote) since November 30, ...
- 18.4k
0
votes
0
answers
643
views
A new generalization of the dimension?
During my research, I came a cross on these notions :
Definition 1: A structure $S$, is a pair $(X, \mathcal T)$ with $X$ a set and $\mathcal T$ a set of subsets of $X$, stable by arbitrary ...
- 4,074
11
votes
1
answer
896
views
Decide if a matrix is transposable
A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations.
Is there an efficient a way/algorithm to decide if a given matrix is
...
- 111
0
votes
1
answer
243
views
A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]
For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define
$A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...
- 1,209
3
votes
0
answers
117
views
Sparsest similar matrix
Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser ...
2
votes
0
answers
60
views
Linearity of a canonical morphism related to scalar extension and coextension
Let $h\colon R\rightarrow S$ be a morphism of commutative ring. Let $M$ and $N$ be $R$-modules. We consider the canonical morphism of $R$-modules $$p\colon{\rm Hom}_R(S\otimes_RM,N)\rightarrow{\rm Hom}...
- 6,700
2
votes
1
answer
3k
views
Lanczos algorithm for finding $k$ smallest eigenvector
I am trying (and have been recommended) to use the Lanczos algorithm to find the $k$ smallest eigenvectors. However, all of the literature seems to talk about this algorithm as a way to estimate the $...
- 21
0
votes
1
answer
3k
views
Cholesky decomposition – non-positive definite matrix
In order to pass the Cholesky decomposition, I understand the matrix must be positive definite. However, I also see that there are issues sometimes when the eigenvalues become very small but negative ...
- 1
3
votes
1
answer
191
views
Hermitian forms over $K\times K$
Let $V$ be a finitely generated module over the ring $R=K\times K$ where $K$ is a field. We fix the switch involution on the ring $R$. Let $H$ be a hermitian form over $V$.
When $V$ is a free module, ...
- 661
1
vote
0
answers
148
views
Traces in associative algebras
Are there some books or papers about the general definition of traces:
If $\mathscr{A}$ is an associative algebra over $K$ then the space of traces is the set of all linear functionals $\tau:\mathscr{...
- 283
12
votes
2
answers
2k
views
Determinant of identity matrix plus Hilbert matrix
I am looking for the determinant
$$ \det(I_n + H_n) $$
where $I_n$ is the $n \times n$ identity matrix and $H_n$ is the $n \times n$ Hilbert matrix, whose entries are given by
$$ [H_n]_{ij} = \frac{...
- 121