New answers tagged linear-algebra
2
votes
Continuous path of unitary matrices with prescribed first column?
Question 1: yes
The map taking a unitary matrix to its first column is a fiber bundle and therefore a fibration.
Question 2: no
Let $n$ be $3$. Make a path $u$ of unitary matrices such that for some $...
- 53.1k
4
votes
Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation
$\det \begin{pmatrix} 1 \end{pmatrix} = 1$ works for any $p$.
$\det \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} = -1$ similarly.
For $n=3$ we require $p \ge 5$. By exhaustion there's no ...
- 6,191
4
votes
Matrices over $\mathbb{F}_p$ that have nonzero determinant under any element permutation
Let's try an easy case: $n=2$. If the matrix has entries $a,b,c,d$ we need $ab - cd$, $ac-bd$ and $ad-bc$ to all be nonzero. In particular if $b,c,d$ are all nonzero there are
at most $3$ forbidden ...
- 53.2k
1
vote
Accepted
Average distance between points of lower dimensional simplices in $\mathbb R^n$
It is highly unlikely that an explicit expression exists.
Even the calculation of the volume of a polytope is a nontrivial problem, solved by Lawrence for simple polytopes. One can possibly use ...
- 110k
5
votes
Sparse representation for continuous function?
Sure, there is a lot about this, but maybe not under that name. You can find related ideas under the name of "best $k$-term approximation", for example in Ron DeVores Acta Numerica paper &...
- 12.2k
2
votes
Does this matrix equation always have a solution?
No. Here is an explicit counter-example for the $i = 3$ case:
$$
A_3^\prime = \begin{bmatrix}
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 & 0 & 0 &...
- 5,513
2
votes
Recover unknown vector through shifted argmax queries
The following argument suggests that the factor $O(n)$ cannot be improved. Suppose for a contradiction that $x\in\{0,1\}^n$ and after queries with $v_1,\dots,v_{n-1}\in\mathbb R^n$ we determined that $...
- 3,937
3
votes
Diagonalizability, orthogonal diagonalizability of higher order tensors and their being or not being dense in some suitable topology
This is indeed true, and this may be the reason why the paper uses decomposable instead of diagonalizable. In the context of general tensors, it may be better to define diagonalizability as $$T = \...
- 296
4
votes
Accepted
On infinity-morphisms between algebras over algebraic operads
It is a typo. The map $f$ should only be assumed to be a morphism of the underlying graded $\mathbb S$-modules.
- 38.8k
2
votes
Accepted
Bounding the size of subspaces of $\mathbb{Z}^n$
Consider the vector $(a,b)\in \mathbb Z^2$ with $a,b$ relatively prime. The vector is orthogonal to $(b,-a)$ and there is no shorter vector with that property. Therefore your constant $C$ must be at ...
- 14.8k
1
vote
Accepted
Is this notion of being "fully" convex closed under set addition?
Have a look at the example from https://mathoverflow.net/a/37683/32507. This gives two convex, closed sets which cannot be separated by linear functionals (even not by discontinuous ones). If I ...
- 1,474
2
votes
Number of Plücker relations for a Grassmannian
The total number of Plücker relations is not the relevant question, since they are not independent; algebraic relations hold between them. In fact on various dense (Zariski) open sets, defined by the ...
- 21
0
votes
Can this system of equations about Newton's formula have concrete result?
Hilarious little detail: the OP assumes that $a\in\mathbb{C}$.
One may wonder if the system is solvable by radicals; then it is easier to assume that $a\in\mathbb{Q}$.
Always there exists a unique -up ...
- 3,442
5
votes
Accepted
Nonempty intersection of cosets of finite-index subgroups
Recall that the intersection of (finitely many) full-rank lattices is also a fumm-rank lattice, as it also has finitely many cosets. So all intersections under consideration, if nonempty, are full-...
- 22k
9
votes
Accepted
Can this system of equations about Newton's formula have concrete result?
Consider $a$ as a transcendental over $\mathbb C$. Then each $x_i$ generates an extension of $\mathbb C(a)$ of degree $n$ whose Galois closure has Galois group the symmetric group $S_n$. Thus once $n\...
- 20.2k
4
votes
Accepted
Conic hull of a rectangle
A counterexample is given by $n=2$, $[a_1,b_1]=[-2,1]$, $[a_2,b_2]=[-1,2]$. (Make a picture.)
Even if the $n$-box $S$ is required to be a subset of $[0,\infty)^n$, the answer will still be no. E.g., ...
- 110k
2
votes
Accepted
Definition and properties of tangent functional
Take any $x\in S$ and $y\in E$. We have to show that
\begin{equation*}
\tau(x,y)\overset{\text{(?)}}=\rho(x,y):=\sup_{f\in T(x)}f(y). \tag{1}\label{1}
\end{equation*}
Generalize the definition of $T(\...
- 110k
4
votes
Existence of a symmetric matrix satisfying certain irreducible conditions
Let $\mathbb{F}$ be a field of characteristic two, and let $K=\mathbb{F}(t)$, the field of rational functions over $\mathbb{F}$. Let $p(x)=x^2-t$. If $A$ is a symmetric matrix with entries in $K$ then ...
- 9,501
2
votes
Theoretical/Practical Implications of DFT Eigenvectors
If you will forgive the self-serving pointer, I wrote a little paper examining this question from the perspective of "what's the sparsest basis of eigenvectors inside the DFT?". The paper's ...
- 3,740
2
votes
Theoretical/Practical Implications of DFT Eigenvectors
A method to construct a real and orthogonal eigenbasis of the DFT matrix has been developed by Dickinson and Steiglitz. There is no explicit closed-form expression, the problem is reduced to the ...
- 172k
Top 50 recent answers are included