Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
10 questions from the last 7 days
-3
votes
1
answer
45
views
Orthogonal Complement of Orthogonal Complement of a Subset [closed]
The following is an attempt at a proof that $S=(S^\perp)^\perp$ for any $S \subseteq V$. I have reason to believe the conclusion is not true in general but I can not find any errors in this proof. If ...
- 1
-3
votes
0
answers
138
views
A presentation for the group $GL(n,\mathbb{Z}_p)$
Let $n\ge 2$. Let $p$ be a prime and $\mathbb{Z}_p$ denote the finite field with $p$ elements.
I want to know about the presentation for the group $GL(n,\mathbb{Z}_p)$ consisting of its generators and ...
- 103
4
votes
0
answers
101
views
Dimension of the intersection of the commuting variety with a particular subspace
Let $\mathcal C$ denote the commuting variety of pairs of matrices in $M_n(\mathbb{C})$, defined as:
$$
\mathcal C = \{ (A, B) \in M_n(\mathbb{C})^2 \mid [A, B] = 0 \}.
$$
It is well known that $\...
- 309
7
votes
1
answer
491
views
Invertibility of a matrix defined using inner product
Let $n,m \geq 1$. We fix $n$ distinct vectors $x_1, ... , x_n \in \mathbb{R}^m$. We define $A \in \mathbb{R}^{n\times n}$ as
\begin{equation}
A_{ij} = x_i^T \left(n x_j - \sum_{1 \leq k \leq n} x_k \...
- 2,306
8
votes
1
answer
358
views
Eigenvalues of a certain combinatorially defined matrix
Let $A_n$ be the matrix whose rows and columns are indexed by pairs
$(i,j)$ with $1\leq i,j\leq n$ and $i\neq j$ (so $A$ is an
$n(n-1)\times n(n-1)$ matrix), whose $((i,j),(k,l))$-entry is 0 if
$i=k$ ...
- 50.8k
0
votes
1
answer
121
views
Inequality for commuting hermitian operators
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $R^n$, $n\geq 2$ (i.e., $p^2_i=p_i=p^*_i$ and $p_1+p_2=\bf{1}$) and $S_1,S_2$ be two commuting hermitian operators on $R^n$ (i.e., ...
- 73
2
votes
0
answers
59
views
Tensor product of two transcendental flat algebras is not a field?
I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
- 173
9
votes
1
answer
291
views
What are the points of the algebra of polynomial functions on an arbitrary vector space?
Let $V$ be an arbitrary vector space over some field $\mathbb{K}$ (UPD: of characteristic 0), $V^*=\mathrm{Hom}(V,\mathbb{K})$ its linear dual. Let $\mathrm{Sym}_\mathbb{K}(V^*)$ be the free ...
- 480
2
votes
1
answer
143
views
Inequality for hermitian matrices
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $\mathbf R^n, n \geq 2$ (i.e., $p_i^2=p_i=p_i^*$ and $p_1+p_2=\bf{1}$) and $S_1, S_2$ be two hermitian operators such that $S_i \...
- 73
4
votes
0
answers
220
views
+50
A question in spin geometry in dimension 8
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ ...
- 954