All Questions
1,923 questions with no upvoted or accepted answers
11
votes
0
answers
683
views
Formula or estimates for $\frac{\operatorname{Tr}(AB)}{\operatorname{Tr}(A)}$
Given two positive definite matrices $A,B$ with nonnegative entries, I seek convenient ways to analytically compute or estimate $\frac{\operatorname{Tr}(AB)}{\operatorname{Tr}(A)}$, where $\...
- 325
11
votes
0
answers
632
views
An elementary linear algebra problem
Let $K$ be a field, and let $E$ be the algebra of $n\times n$ matrices over $K$. Let $V_0$ and $V_1$ be the (left) $E$-modules of matrices of size $n\times n_0$ and $n\times n_1$. Let $W \subseteq V_0$...
- 4,015
11
votes
0
answers
305
views
Generalized Classical Adjoints and Factorizations of the Characteristic Polynomial
This is idle noodling, and I'm prepared to learn that it's foolish as well as idle. But....
Let $M$ be an $n\times n$ matrix over, oh, let's say an algebraically closed field for now. There have ...
- 23k
11
votes
0
answers
576
views
What's known about the mod 2 reduction of the level l Jacobi modular equation?
Motivation:
Let $\ell$ be an odd prime. Let $A$ in ${\mathbb Z}/2[[x]]$ be $x+x^9+x^{25}+x^{49}+...$, and $B=A(x^\ell)$. One can use the level $\ell$ Jacobi modular equation to get a polynomial ...
- 5,422
11
votes
0
answers
1k
views
Do the Standard Conjectures imply parts of the "Weil II" Riemann Hypothesis?
It is known that Grothendieck's Standard Conjectures on algebraic cycles imply the Riemann Hypothesis of the original Weil Conjectures. However, do they also say something about the version of the ...
- 1,764
11
votes
1
answer
2k
views
Quantifying the failure of the Cholesky factorization test for indefinite matrices
The Cholesky factorization is the classic test to check if a matrix is positive definite. In infinite precision it is also an exact test: A matrix has a Cholesky factorization iff it is positive ...
- 3,217
10
votes
0
answers
371
views
How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?
$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
- 1,335
10
votes
0
answers
399
views
Words and ranks
Let me state two problems that look very much alike. The first one can be solved putting together answers that different people have given to some questions I asked here a few weeks ago. The second ...
- 20.2k
10
votes
0
answers
312
views
Triangle $X'\to X\to X''\to\Sigma X'$ splits if $X\simeq X'\oplus X''$?
Given a commutative ring $R$ and a distinguished triangle $X'\to X\to X''\xrightarrow e\Sigma X'$ in the derived category $D(R)$, where $X',X,X''$ are perfect complexes. If we have an equivalence $X\...
user20948
10
votes
0
answers
393
views
Interpretation of determinants on commutative rings
In real Euclidian space, the result of the determinant can be interpreted as the oriented volume of the image of the unit cube under an invertible linear map.
This interpretation conceptually depends ...
- 323
10
votes
0
answers
238
views
Generalized eigen property of a matrix
Given a $n \times n$ invertible matrix $A$, I am interested in the set
$$
\mathcal{S}(A) = \{ D \textrm{ diagonal matrix } \mid \det(D - A) = 0 \}.
$$
Thus, for all eigenvalues $\lambda_i$, we have $...
- 909
10
votes
0
answers
420
views
Gram matrix determinant in dimension 4 and $E_8$
Consider a determinant of a Gram matrix in dimension $4$.
$$\begin{vmatrix}
1 & -\cos(\alpha_1) & -\cos(\alpha_2) & -\cos(\alpha_3)\\
-\cos(\alpha_1) & 1 & -\cos(\alpha_6)& -\...
- 4,316
10
votes
0
answers
229
views
Maximum dimension of a space of $n\times n$ real matrices with at least $k$ nonzero eigenvalues
Let $M_n(\mathbb{R})$ denote the $n^2$-dimensional real vector space
of real $n\times n$ matrices. Let $\rho_k(n)$ denote the maximum
dimension of a subspace $V$ of $M_n(\mathbb{R})$ such that every
...
- 50.8k
10
votes
0
answers
225
views
Cospectral mate of rhombic dodecahedron
I am wondering if the following pair of cospectral graphs was previously known.
The rhombic dodecahedron graph looks like this (graph6 string: 'M?????rrAiTOd_YO?'):
As far as I know, it was previously ...
- 2,339
10
votes
0
answers
952
views
Dimensions of dual vector spaces
Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
- 18.7k
10
votes
0
answers
277
views
A $k \times n$ matrix with a lot of invertible $k \times k$ submatrices over $\mathbb{F}_2$
In the appendix of the paper by Tolhuizen (
http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=841182) there is a very fast and easy probabilistic proof that for $k=cn$ where $c \in [0,1]$ is fixed, ...
- 3,004
10
votes
0
answers
477
views
Name for an operation on matrices?
Given two matrices $A$ and $B$ of size $a \times n$ and $b \times m$ consider the following operation $A \dagger B$ whose result is an $a b^n \times n m$ matrix. $A \dagger B$ is a block matrix with $...
10
votes
0
answers
323
views
The mod 3 reduction of some powers of delta
Let f in Z/3[[x]] be the mod 3 reduction of the Fourier expansion of the normalized weight 12 cusp form delta for the full modular group. The exponents appearing in f are all 1 mod 3. Fix
k>0 and ...
- 5,422
9
votes
0
answers
378
views
How many orthogonal matrices (not orthonormal) are there with entries in $\{0,1,−1\}$?
Here by orthogonal matrix I mean just the rows are mutually orthogonal. Two such matrices are equivalent if one can be obtained from the other by permutations of rows and columns, or change of signs ...
- 745
9
votes
0
answers
360
views
Factorisation of a polynomial from the Boolean algebra
Let $B_n$ denote the Boolean algebra of a set with $n \geq 2$ elements and $C_n$ the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else, where $i,j\in B_n$.
Let $M_n:=C_n+C_n^T$ and $...
- 26.5k
9
votes
0
answers
270
views
The number of non-singular $n\times n$ matrices over $\mathbb{F}_2$ with exactly $k$ non-zero entries
Suppose $M_{n}^{k}$ is the number of non-singular $n\times n$ matrices over $\mathbb{F}_2$, that have exactly $k$ non-zero entries.
Is there some sort of formula to calculate $M_n^k$?
If $k < n$ ...
- 2,618
9
votes
0
answers
261
views
Does there exist an algorithm that decomposes a matrix into a minimal number of elementary matrices for $F_{2}$?
If $i \neq j$, then let $C_{i,j} : F_{2}^{n} \to F_{2}^{n}$ be the elementary linear transformation defined by
$$C_{i,j}(x_{1},\dots,x_{i},\dots,x_{j},\dots,x_{n}) :=(x_{1},\dots,x_{i},\dots,x_{i}\...
9
votes
0
answers
625
views
Eigenvalues of leading principal submatrix of the Clement-Kac-Sylvester tridiagonal matrix
It's well-known that the eigenvalues of the Clement-Kac-Sylvester tridiagonal matrix
$$\begin{pmatrix}
0 & n-1 & 0 & \dots & 0 \\\
1 & 0 & n-2 & \dots & 0\\\
0 & ...
- 139
9
votes
0
answers
360
views
Finding $U,V$ in Thompson's Formula
Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that:
$e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$
Given $a,b \in \mathfrak{su}(4)$ defined by:
$a=J_x ...
- 2,099
9
votes
0
answers
256
views
Intersection of Springer fibre and Schubert cell
Let us consider intersections of Springer fibres and Schubert cells in type A.
Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let
$$
F_Y = \{ V_0 = 0 \subset V_1 \subset \...
- 4,612
9
votes
0
answers
980
views
Strong convexity of the trace of the square root of a matrix function
Any clues about how to prove that the following function is strongly-concave in $x$? (We conjecture it is $2$-strongly concave but cannot prove it. We have already proved strict concavity through ...
- 91
9
votes
0
answers
434
views
When is a product of hyperbolic matrices hyperbolic?
Suppose $A_1,\ldots,A_n$ is a sequence of $2 \times 2$ complex matrices such that $| \det(A_j) | =1$ and $ | \mathrm{tr}(A_j) | > 2 $ for each $j$. What kinds of reasonable restrictions can one ...
- 437
9
votes
0
answers
464
views
An identity for Hankel determinants
Is the following result about Hankel determinants known or a simple consequence of some known results?
Let
$f(x) = \frac{\displaystyle 1}{{\displaystyle 1 - \frac{{a x^{m + 2}}}{\displaystyle {1 - \...
- 5,622
9
votes
0
answers
560
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
- 5,422
9
votes
0
answers
1k
views
Ample vector bundles, $H^1=0$ and global generation in characteristic $p$
This is a follow up from Ample vector bundles on curves question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective ...
- 2,976
9
votes
0
answers
1k
views
coordinate-free proof of transitivity of norms or traces
Hello:
Suppose $A$ is a finite free $B$-algebra and $B$ is a finite free $C$-algebra. Does anyone
know a coordinate-free proof (i.e. without choosing bases) of the identity:
$N_{A/C} = N_{B/C}\circ ...
- 647
8
votes
0
answers
172
views
Random walk on matrix until singularity
Consider a random walk on matrices, where one starts with the matrix $M=I_n$ and at each step randomly chooses an entry of $M$ to increase by $1$.
I’m interested in two things about this walk:
What’s ...
- 541
8
votes
0
answers
337
views
How to check two matrices for similitude over $\mathbb{Z}$?
General question. Let $A$ and $B$ be two $n\times n$-matrices over
$\mathbb{Z}$. How do I algorithmically check whether $A$ and $B$ are similar
(i.e., conjugate in the ring $\mathbb{Z}^{n\times n}$)?
...
- 33.8k
8
votes
0
answers
232
views
Decay of orthogonal contributions in a random set of vectors
Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\...
- 2,252
8
votes
2
answers
414
views
Recovering eigenvalues of a matrix from its $p$th compound matrix
This question was motivated by Find the determinant of a matrix given the determinant of all $p\times p$ sub-matrices?. Let $A$ be an $n\times n$ matrix over a field. Suppose
we are given the $p$th ...
- 50.8k
8
votes
0
answers
285
views
Matrix decompositions as monoid isomorphisms. Ever considered before?
I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question:
...
- 4,943
8
votes
0
answers
293
views
Image of multiplication map in tensor powers of finite-dimensional ring
Let $R$ be a (commutative, unital) ring of dimension $n$ over a field $k$. Assume the characteristic of $k$ is greater than $n$.
Then $R^{\otimes n}$ has a natural ring structure, together with an $...
- 149k
8
votes
1
answer
531
views
How large can the dimension of a 'Span of powers of a finite field basis' be?
Let $p$ be a prime. For finite field $\mathbb{F}_{p^k}$ and $d\in\mathbb{Z}^+$, I am considering the following quantity, where we interpret the field $\mathbb{F}_{p^k}$ also as a $\mathbb{F}_p$-vector ...
- 89
8
votes
0
answers
333
views
Monotonicity of log determinant of Gaussian kernel matrix
Let \begin{equation} k({x},{y}) = \sigma \exp\left(-\frac{(x-y)^2}{2\theta^2}\right)\end{equation}
be a squared-exponential (Gaussian) kernel, with $\sigma,\vartheta>0$. Consider, for a set of $N$ ...
- 55
8
votes
0
answers
300
views
Which algebra is $\mathbb{F}_p \otimes_{HH^{\cdot} (\mathbb{F}_p)} \mathbb{F}_p$?
Let $HH^{\cdot}(\mathbb{F}_p)$ be the Hochschild cohomology of $\mathbb{F}_p$ over $\mathbb{Z}$, which as a $E_1$ ring is simply $\mathbb{F}_p[x]$ with $x$ in cohomological degree $2$. Then it's clear ...
- 2,356
8
votes
0
answers
688
views
An Azumaya algebra from a vector bundle, and a construction of Belov-Kanel and Kontsevich
Let $S/k$ be a scheme over a perfect field $k$ of characteristic $p>0$.
In Automorphisms of the Weyl Algebra, Belov-Kanel and Kontsevich write down the map
$$\alpha: H^0(\Omega^1_{S/k}/d\mathcal O) ...
- 2,368
8
votes
0
answers
219
views
Elementary proof of an inequality for the Radon-Hurwitz numbers
Edit:
In all likelihood, the original question does not have a positive answer (see comment by abx).
Modified question: Let $\rho_H(n)$ be the maximal dimension of a space of symmetric real matrices ...
- 338
8
votes
0
answers
157
views
Periods of Coxeter transformation associated to root posets
$\DeclareMathOperator\Co{Co}$Let $P$ be the root poset associated to a simple Lie algebra.
Let $L=L(P)$ denote the distributive lattice of order ideals of $P$ and let $\Co_L$ denote the Coxeter matrix ...
- 26.5k
8
votes
0
answers
413
views
Eigenvalues of cyclic stochastic matrices
Let's consider the following $n \times n$ cyclic stochastic matrix
$$ M= \begin{pmatrix}
0 & a_2 & & & &b_n \\\
b_1 & 0& a_3& &&& \\\
& b_2 & ...
- 181
8
votes
0
answers
194
views
Geometric mean of three or more positive definite matrices
The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...
- 13.4k
8
votes
0
answers
233
views
A conjecture on simplex
Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$
Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{...
- 705
8
votes
0
answers
577
views
A rank inequality
Suppose
$$M := \begin{bmatrix}
M_{11} & \cdots &M_{1d} \\
\vdots & \ddots & \vdots \\
M_{d1} & \cdots & M_{dd}
\end{bmatrix}$$
is a $d \times d$ block matrix such that
$$M_{...
- 500
8
votes
0
answers
492
views
Strange determinant inequality $\det(C+ xA) \det(C-xA) \le (\det C)^2$
Let $A$ be an all-one $3$-by-$3$ matrix, let $C$ be a $3$-by-$3$ matrix, and let $x$ be a real number. How might one prove the following inequality?
$$\det(C+ xA) \det(C-xA) \le (\det C)^2$$
- 99
8
votes
0
answers
471
views
Sheaf whose singular support is not Lagrangian
For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the ...
- 18.7k
8
votes
0
answers
298
views
If $A$ is an algebra, $Sym^n(A)$ is an algebra. Where can I learn more about this algebra structure?
$\newcommand{\Vect}{\mathsf{Vect}}
\newcommand{\nats}{\mathbb{N}}
\newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\Alg}{\mathsf{Alg}}
\newcommand{\CAlg}{\mathsf{CAlg}}
\newcommand{\Hom}{\mathrm{Hom}}$
Let ...
- 64k