All Questions
Tagged with linear-algebra ag.algebraic-geometry
31 questions
2
votes
1
answer
184
views
Count N-tuples of commuting matrices over $F_q$ is given by polynomials with pattern $\sum q^{A_i(N)} P_{i}(q) $, where $P_i$ - do not depend on $N$?
Count pairs of $k \times k$ commuting matrices over finite field $F_q$ is given by certain polynomials in $q$ (which is quite rare phenomena for algebraic varieties) and have interesting generating ...
- 24.9k
54
votes
5
answers
2k
views
Unusual symmetries of the Cayley-Menger determinant for the volume of tetrahedra
Suppose you have a tetrahedron $T$ in Euclidean space with edge lengths $\ell_{01}$, $\ell_{02}$, $\ell_{03}$, $\ell_{12}$, $\ell_{13}$, and $\ell_{23}$. Now consider the tetrahedron $T'$ with edge ...
- 10.1k
12
votes
1
answer
5k
views
Closest 3D rotation matrix in the Frobenius norm sense
Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm:
\begin{equation}
\|R-M\|_F
\end{equation}
Is there a closed form solution for $R$, or is it ...
- 561
7
votes
2
answers
4k
views
Number of Plücker relations for a Grassmannian
Is it true that the number of Plücker relations for a Grassmannian $Gr(k,n)$ is equal to the dimension $k(n-k)$ of said Grassmannian? So far, for $Gr(2,5)$, I get exactly five Plücker relations: $$p_{...
- 349
47
votes
4
answers
7k
views
Using linear algebra to classify vector bundles over ℙ¹
There is a theorem of Grothendieck stating that a vector bundle of rank $r$ over the projective line $\mathbb{P}^1$ can be decomposed into $r$ line bundles uniquely up to isomorphism. If we let $\...
- 533
36
votes
3
answers
2k
views
Are large powers of polynomials linearly independent?
Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
- 4,653
16
votes
4
answers
3k
views
How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 1,795
11
votes
2
answers
10k
views
Derivative of eigenvectors of a matrix with respect to its components
Suppose that $B$ is a real, positive-definitive symmetric ($3\times3$) matrix (more accurately, $B$ is a tensor) with distinct eigenvalues, and that we can write it as
$$
B= \sum_{i=1}^3 \lambda_{i}(...
- 221
3
votes
1
answer
399
views
Maximal common isotropic subspace for a finite family of skewforms
Let $V$ be a vector space of dimension $n$ over a field $F$. An alternating bilinear form $\alpha\colon V \times V \rightarrow F $ will be called a skewform. A subspace $W$ is isotropic for $\alpha$...
- 923
3
votes
2
answers
452
views
Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture
Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
- 24.9k
44
votes
2
answers
2k
views
Is this lemma in elementary linear algebra new?
Is anyone familiar with the following, or anything close to it?
Lemma. Suppose $A$, $B$ are nonzero finite-dimensional vector spaces
over an infinite field $k$, and $V$ a subspace of $A\otimes_k B$
...
- 595
14
votes
0
answers
602
views
Is the Zariski density proof of Cayley-Hamilton circular?
This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). ...
- 118k
13
votes
4
answers
3k
views
Multivariate analogue of Vandermonde determinant
Dear all,
Consider the $(n+1)\times (n+1)$ matrix $A$ with indeterminates $X_i, Y_i$, $0\leq i\leq n$ such that the $(i,j)$-th entry is given by $X_i^jY_i^{n-j}$. The $i$-th row is $(X_i^n,X_i^{n-1}...
- 537
9
votes
3
answers
696
views
I want to find a smooth section of the map from the Stiefel manifold to the Grassmanian manifold
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 155
9
votes
1
answer
1k
views
How to write down the determinant of a quasi-isomorphism?
This question about the determinant of a perfect complex reminded me of an old question that I had.
The construction of the determinant (as in MR1914072 or MR0437541) is a difficult piece of ...
- 3,284
7
votes
1
answer
355
views
Injectivity of matrix "fingerprint"
Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries.
For any matrix $A$, define
$$ D_A(i,j)=\...
- 3,979
6
votes
1
answer
778
views
Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$
I need some help about the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ E(f):=\sum_{j=1}^{...
- 307
6
votes
4
answers
658
views
Reference for an algebraic group preserving a cubic form
Let $R=k[u,v,w]$ and $p\in R$ be a cubic form. Let $G$ be the group of graded automorphisms of $R$ which preserve $p$, i.e., $G$ is the subgroup
of $GL_3(k)$ consisting of elements $g$ such that $g(p) ...
- 63
6
votes
2
answers
1k
views
Systems of simultaneous real quadratic equations
Starting from a problem in spectral graph theory, I got dragged into a problem in combinatorial matrix theory about constructing $n\times n$ real orthogonal matrices with a specified pattern of zero/...
6
votes
3
answers
2k
views
Finding the action of the symplectic group on the Siegel-half plane
Let $S$ be the Siegel-half plane of dimension $n$, i.e. the set of complex $n \times n$ matrices $Z$ which are symmetric and whose imaginary part is positive-definite. In dimension 1 we can identify $...
- 4,343
5
votes
2
answers
1k
views
Lie's theorem in characteristic $p$
Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...
- 3,741
5
votes
2
answers
707
views
Explicit equation of Dickson invariant / quasideterminant / special orthogonal group over the integers
Consider $2n$ coordinates $x_1,\ldots,x_n,y_1,\ldots,y_n$ and the quadratic form $q = \sum_{i=1}^n x_i y_i$. Now call $O(q,A)$ (orthogonal group of $q$) the group of $(2n)\times(2n)$ matrices, with ...
- 32.5k
4
votes
4
answers
3k
views
The multiplicity of the max eigenvalue in matrix multiplication
Suppose that eigenvalues of two real square matrix $A$ and $B$ are $1 = \lambda^A_1 > \lambda^A_2 \geq \ldots \geq \lambda^A_n > 0 $ and $1 = \lambda^B_1 > \lambda^B_2 \geq \ldots \geq \...
- 41
3
votes
0
answers
234
views
Hurwitz–Radon problem for $ \mathbb{Q} ^{n} $
What is the maximal number of orthogonal operators $ A _{1} , \dotsc, A _ {m} $ in $ \mathbb{Q}^{ n } $ satisfying the relations $ A_{i}^{2} = - I $ and $ A_{i}A_{j} + A_{j}A_{i} = 0 $ for $ i \neq ...
- 923
3
votes
1
answer
389
views
Dimension of hermitian rank at most $k$ matrices over quaternions
In a (right) finite dimensional quaternionic Hilbert space there is an analogue of the spectral theorem (see theorem 4.6 in Farenick and Pidkowich) for normal matrices in $\mathbb{H}^{m\times m}$, ...
- 3,209
2
votes
2
answers
2k
views
Constant row-column sum matrices?
Given an integer $T$ how many $n\times n$ matrices in $\mathbb Z_{\geq0}^{n\times n}$ we have such that every row and column sum to same $T$?
Do the set of constant row and column sum matrices form ...
- 13.9k
2
votes
0
answers
136
views
Linear independence of functions
Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
- 859
2
votes
1
answer
927
views
Paralel bezier curve
If I have a cubic Bezier curve specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined the ...
- 123
1
vote
1
answer
367
views
Irreducibility of a resultant of real and imaginary parts of a characteristic polynomial
The following question is motivated by the study of a stability border for a robust linear time-invariant control system.
Let us we have an affine family of $n\times n$ matrices with indeterminate ($\...
- 413
0
votes
1
answer
311
views
Intuitive explanation of concentration of the measure for spheres [duplicate]
What is the concentration of the measure(c.o.m.)?
I am struggling with the following sentence;
"The phenomenon of the concentration of the measure for spheres in dimensions larger than 2."
I tried ...
- 11
0
votes
1
answer
154
views
Dimension of a similarity class
Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...
- 3,741