## 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 $...

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 ...

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 ...

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 ...

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 &...

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 &...

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
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 = \...

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.

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 ...

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 ...

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 ...

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 ...

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-...

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\...

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., ...

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(\...

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 ...

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 ...

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 ...

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