All Questions
15 questions
6
votes
1
answer
239
views
Attempts to define a matrix exponential over (as much as possible) general fields
Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as
$$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$
where ...
- 361
1
vote
2
answers
333
views
Condition for equality of modules generated by columns of matrices
Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
- 431
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
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
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
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
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
10
votes
1
answer
537
views
Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
- 6,349
4
votes
1
answer
787
views
What is a degenerate Legendre Transformation?
I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...
- 245
27
votes
6
answers
6k
views
Origin of exact sequences
I have seen exact sequences appearing a lot in algebraic texts with different purposes. But I've never seen names of the people associated with it. Also I don't understand what's so good about showing ...
- 443
3
votes
1
answer
3k
views
Diagonalize the simultaneous matrices and its background [closed]
For two $n \times n$ nonnegative definite Hermitian matrices $A$ and $B$ over the real number field $\mathbb R$:
Question1:Is there always a
nonsingular matrix $P$ over the same
field $F$ which ...
- 8,071
5
votes
0
answers
442
views
A reference on semisimple linear algebra
Is there any literature where the tools familiar from (multi)linear algebra are systematically transferred to the setting of semisimple modules over noncommutative rings?
In fact this question is a ...
- 4,010
5
votes
3
answers
1k
views
adjoint of multiplication operator in a commutative algebra
Dan Popescu asked me the following question, and since I'm not an expert I'm throwing his question on MO.
Suppose that $A$ is a finite-dimensional vector space over an ordered field $k$ with $\...
- 6,614
1
vote
0
answers
396
views
Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.
I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and $y,z \in \mathbb{R}^n$ are vectors. I also need to consider ...
- 4,922