All Questions
1,809 questions with no upvoted or accepted answers
4
votes
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101
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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
4
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0
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221
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+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
4
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0
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103
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Relationship between characteristic polynomials of a matrix and its adjoint representation
Let $A \in \mathrm{M}_n(F)$ be a matrix over a field $F$. Consider its adjoint representation $\mathrm{ad}_A \in \mathrm{End}(\mathrm{M}_n(F))$, defined by
$$
\mathrm{ad}_A(X) = [A, X] = AX - XA.
$$
I ...
- 309
4
votes
0
answers
291
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Lower bound on size of the set of sums and differences of non-orthogonal pairs of vectors over finite field
Consider $\mathbb{Z}_m^n$, an $n$-dimensional vector space over $\mathbb{Z}_m$.
For two sets of vectors $P = \left\{ p^i \right\}$ and $Q = \left\{ q^j \right\}$ and a skew-symmetric matrix $S_{ij}=\...
- 91
4
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0
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284
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Institutional approach to linear algebra
In Diaconescu's book Institution Independent Model Theory, it is mentioned on p. 37 that linear algebra can be viewed as an institution. Specifically, we have the following
Definition. An institution ...
- 10.1k
4
votes
0
answers
87
views
Non-zero element in diagonal of cofactor matrix of symmetric 0-1 matrix with non-zero determinant, zero diagonal and odd number of rows
Is the following statement correct?
Let $A$ be a symmetric 0-1 matrix with non-zero determinant, all diagonal elements equal to 0 and an odd number of rows. The diagonal of the cofactor matrix of $A$ ...
- 41
4
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0
answers
108
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Larger possible chain of closed subspaces in the dual of a Banach space
In this question, is demonstrated that a separable space can have a chain (ordered by inclusion) of closed subspaces with uncountable many subspaces.
My question is the following. If $X$ is an ...
- 153
4
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0
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238
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Conjectural values of some determinants involving Legendre symbols (I)
$\newcommand\Legendre{\genfrac(){}{}}$Let $p$ be an odd prime, and let $\Legendre\cdot p$ be the Legendre symbol. In 2003, Robin Chapman evaluated the determinants
$$\det\left[\Legendre{i+j}p\right]_{...
- 15.6k
4
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0
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99
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If matrices $A$ and $B$ are normal with $\sigma(A),\sigma(B)\subseteq \mathbb{R}\cup \mathbb{T}$, does $\text{rank}([A,B])=\text{rank}([A^*,B])$?
Here $\mathbb{T}=\{z\in\mathbb{C}: |z|=1\}$ denotes the unit circle in the complex numbers.
This holds, if we have $\sigma(A)\subseteq \mathbb{R}$ or $\sigma(A)\subseteq \mathbb{T}$ (independent of $B$...
- 151
4
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0
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140
views
Does an instance of this generalisation of the determinant exist?
Let $n$ be composite, $d$ a divisor greater than $1$ and $m=n/d$. Does anybody know if there is a general mapping $T$ from $n×n$ matrices to $m×m$ matrices that preserves the determinant? Over a field ...
- 1,245
4
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0
answers
97
views
The spectra of Hodge-Laplace operators
If we have a sequence of linear maps and finite dimensional inner product spaces $$X\xrightarrow{f} Y\xrightarrow{g}Z$$ such that $g\circ f=0$, then we can consider the Hodge-Laplace operator $$\Delta:...
- 47.3k
4
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1k
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Number of arrangements that contain at least 1 path from top to bottom of 2D matrix
I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white.
With that information, I can easily calculate the total number of black element arrangements that exist ...
- 47
4
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0
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262
views
Two questions about three circulant matrices
Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$
$$2AA^T+BB^T+CC^T=(4n+4)I-4J$$
where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
- 696
4
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219
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Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$
EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect.
Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
- 7,300
4
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147
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Frobenius norm bounds on exponentials of anti-Hermitian matrices
Suppose $X$ and $Y$ are two anti-Hermitian matrices satisifying $\|X\|, \|Y\| \leq \pi$, where $\|\cdot\|$ is the spectral norm. I'm trying to prove the following bounds on the Frobenius norm of the ...
- 41
4
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answers
164
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Dimensionality reduction preserving cyclic traces
Suppose that I have $n$ matrices $A_1, \ldots, A_n \in \mathbb{R}^{m \times m}$ with $m \gg n$. Can I find $n$ new matrices $B_1, \ldots, B_n \in \mathbb{R}^{n \times n}$ that have the same 3-way ...
- 777
4
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0
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181
views
Fuss-Catalan: how does equality of these determinants hold?
There are many ways that the Catalan numbers seemed to have been generalized, one among them is through what Graham-Knuth-Patashnik (in Concrete Mathematics) dubbed as the Fuss-Catalan numbers
$\frac1{...
- 43.2k
4
votes
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216
views
Is an orthogonal direct sum decomposition with respect to two quadratic forms necessarily unique up to isomorphism
Consider two quadratic forms $Q$ and $P$ over a finite dimensional vector space $V$ over a quadratically closed (or perhaps Pythagorean) field $F$. If $V$ can be decomposed as $V = V_1 \oplus V_2 \...
- 4,943
4
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98
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A question on products of linear combinations of complex matrices
Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds
$$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{...
- 69
4
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answers
134
views
Irreducibility of polynomials associated to binomial coefficients
Let $n \geq 2$.
Let $M_n$ be the $(n+1) \times (n+1)$ matrix with entries $\binom{l}{k}$ for $0 \leq l,k \leq n$ and $U_n=M_n + M_n^T$ and let $f_n(x)$ denote the characteristic polynomial of $U_n$.
...
- 26.5k
4
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0
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108
views
Doubly stochastic matrices that remain doubly stochastic after conjugating by the character table of a finite abelian group
I am curious if anything is known about the following.
Let $\Gamma$ be a finite abelian group, and let $\chi$ be its character table, normalized so that it is a unitary matrix. E.g., if $\Gamma$ is $\...
- 2,339
4
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112
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Duality for finite quotient groups of finitely generated free abelian groups
$\newcommand{\Z}{{\Bbb Z}}
\newcommand{\Q}{{\Bbb Q}}
\newcommand{\Hom}{{\rm Hom}}
$ The following lemma is certainly known.
Lemma (well-known).
Let $B$ be a lattice (that is, a finitely generated ...
- 14.1k
4
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answers
161
views
Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$
$\newcommand{\GL}{\operatorname{GL}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\C}{{\Bbb C}}
$For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle
of $G=\GL_{n,\R}\,$, that is,
an invertible ...
- 14.1k
4
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0
answers
107
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Two questions about $\mathbf{PSL}(V)$ with $V$ a vector space over a division ring
Let $V$ be a (possibly infinite-dimensional) vector space over a division ring $d$, and consider the projective special linear group $\mathbf{PSL}(V)$. We suppose that if the dimension of $V$ would be ...
- 4,547
4
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989
views
Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries
Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...
- 41
4
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0
answers
148
views
Derivative of characteristic polynomial of a graph and derivative of characteristic polynomial of a vertex-deleted subgraph have a common root
Let $G$ be a simple graph and $G-i$ be one of its vertex-deleted subgraphs. Let $\phi(G,x)$ and $\phi(G-i,x)$ be the characteristic polynomials of $G$ and of $G-i$ respectively, with respect to their ...
- 327
4
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0
answers
207
views
Isomorphism between tensor product of exterior power spaces
Suppose that $V_1, V_2, V_3$ are finite dimensional vector spaces over $\mathbb{C}$ of dimensions $d_1, d_2, d_3$, respectively. Suppose that $V_1, V_2, V_3$ are equipped with inner products, so that ...
- 730
4
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211
views
Diagonalization over valuation rings
Let $\mathcal{R}$ be a valuation ring, and consider an $\mathcal{R}$-linear endomorphism $L:\mathcal{R}^{n}\rightarrow \mathcal{R}^{n}$. Is there any criterion for telling when $L$ can be diagonalized?...
- 541
4
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311
views
Estimates of the Frobenius norm of commutator
Let $A,B$ be two unitary matrices in $U(n)$, and $\|\cdot\|_{F}$ denote the Frobenius norm (or Hilbert Schmidt norm on the finite dimensional $M_n(\mathbb{C})$). I am looking for estimates of the ...
- 949
4
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0
answers
211
views
Conjecture on convergence of iterated near-matrix square root
Here is a simple problem that has stumped me for some time; sharing with the community, as I suspect it has been solved somewhere, or is immediately implied by the correct theorem.
Let $\textbf{diag}: ...
- 193
4
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0
answers
196
views
What is the minimum nonzero rank in a random subspace of matrices?
Fix positive integers $m$, $n$, and $k\leq mn$, and draw a $k$-dimensional subspace $S\leq\mathbb{R}^{m\times n}$ uniformly from the Grassmannian.
What is known about the random variable
$R(m,n,k):=\...
- 7,633
4
votes
1
answer
228
views
A question on eigenvalue of parametric matrix
Is there a way to efficiently check if all matrices in the following set are Hurwitz stable (eigenvalues strictly in the left-hand plane)?$$\left\{ A \in \Bbb R^{n \times n} : \ell_{i,j}\leq A_{i,j} \...
- 1,216
4
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0
answers
163
views
Matrix logarithm of unitary factor from polar decomposition of product of positive definite matrices
This question is crossposted from Math Stackexchange here. I crosspost without much edits as I think this is the best way to phrase the question and because I received no feedback on the original post ...
- 41
4
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answers
113
views
Index of norm $ 1 $ subgroup in a cyclic extension
Let $L/\mathbb{Q}$ be a cyclic galois extension of degree $ 2n $ and $\sigma $ be a generator of $\operatorname{Gal}(L/\mathbb Q)$. Let $ U $ be the collection of all norm $ 1 $ elements of $L^\times$...
- 923
4
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0
answers
152
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How to show the set of stable polynomials equals to the set of Lorentzian polynomials in degree 2
Give a homogenous polynomial $f\in \mathbb{R}[x_1,\dots,x_n]$ of degree $2$ in $n$ variables, we can consider $f$ as a quadratic form.
We call $L_n^2:=$ the set of quadratic forms with nonnegative ...
- 41
4
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0
answers
259
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On the characteristic polynomial of the Vandermonde matrix
Let $A_n$ be the $n \times n$-Vandermonde matrix (see for example https://en.wikipedia.org/wiki/Vandermonde_matrix )viewed as a matrix over the fraction field of the polynomial ring over a field $K$ (...
- 26.5k
4
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244
views
Free vector space on a filtered limit
$\DeclareMathOperator\Set{Set}\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Coalg{Coalg}\DeclareMathOperator\ProVect{ProVect}\DeclareMathOperator\prolim{prolim}
$Let $K$ be a field and $F: \Set \...
- 1,361
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0
answers
176
views
Finding roots of equation with gamma functions
Encountered this function in one of my research problems
$$\frac{\Gamma \left(1-\dfrac{i c}{a}-\gamma \right) \Gamma \left(1+\dfrac{i c}{a}+\dfrac N 2-\gamma \right)}{\Gamma \left(1+\dfrac{i c}{a}-\...
- 199
4
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0
answers
75
views
All possible discrete probability distributions arising from a finite length product of stochastic matrices
Consider a discrete probability distribution $x = (x_1,\ldots,x_n)$, where $x_i\ge0$, $\sum_ix_i=1$, and a set of $M$ stochastic matrices $P^1,\ldots,P^M\in\mathbb{R}^{n\times n}$, where all $P_{ij}\...
- 179
4
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281
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Classification of special symmetric Frobenius algebras over real vector spaces
Is there a general classification of special symmetric Frobenius algebras over real vector spaces? I know that $n\times n$ matrix algebras, the quaternions, the complex numbers, the trivial algebra, ...
- 3,001
4
votes
0
answers
102
views
Incidence relations of subspaces with infinite descending flags
Let $W = \prod_{k \in \mathbb N} V_k$ be an infinite product of vector spaces, and let $V = \oplus_{k \in \mathbb N} V_k$ be the corresponding sum. Already the case where $V_k$ is 1-dimensional for ...
- 63.9k
4
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0
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144
views
Linearly dependent vector fields not spanned by fewer continuous vector fields
This is a follow-up to the question https://math.stackexchange.com/questions/3845080/linearly-dependent-vector-fields-that-are-not-spanned-by-fewer-continuous-vect, which attracted very little ...
- 141
4
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0
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1k
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Reference for matrices with all eigenvalues 1 or -1
In a homological algebra problem I am in the situation that I have an invertible (over $\mathbb{Z}$) integer matrix $X$ and a permutation matrix $Y$ such that $N:=XY$ is a matrix with all eigenvalues ...
- 26.5k
4
votes
0
answers
124
views
Distributive lattices with periodic Coxeter matrix
Let $L$ be a finite distributive lattice and $U$ its incidence matrix with entries $u_{i,j}=1$ iff $i \leq j$ and $u_{i,j}=0$ else.
Then $U^{-1}$ is the Moebius matrix of $L$ and $C_L:=- U^{-1} U^{T}$ ...
- 26.5k
4
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0
answers
1k
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Can an orthogonal matrix move monotonically toward a signed permutation matrix?
The question is motivated by this question on Mathematics SE.
Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
- 209
4
votes
0
answers
109
views
Characterization of "PSD-Squared" Matrices
$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$This question can be thought of as an offshoot of this MO question from a few months ago. Let $M_n(\mathbb{C})$ denote the set of $n \times n$ ...
- 6,351
4
votes
0
answers
457
views
Inequalities for trace/eigenvalues of product of multiple 2x2 matrices
Consider the matrix product $\prod_i^n A_i$,
where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
- 695
4
votes
0
answers
223
views
$\epsilon$-net under Hausdorff distance
Consider linear subspaces of $\mathbb{R}^n$. For two subspaces $X$ and $Y$, we define their Hausdorff distance as
$$
{\displaystyle d_{\mathrm {H} }(X,Y)=\max \left\{\,\sup _{x\in X, |x|_2=1}\inf ...
- 1,503
4
votes
0
answers
247
views
Eigenvalues of structured matrices
Let $A=(a_{i,j})$ be an $n\times n$ matrix with $a_{j,j+1}>0,\; 1\leq j\leq n-1,$ and $a_{j,j-2}>0,\; 3\leq j\leq n$, the rest of the entries are zeros.
Is the following fact known:
All ...
- 91.8k
4
votes
0
answers
144
views
A Pythagorian inequality characterization of inner-product spaces
Let $(X,\|\cdot\|)$ be a real normed space. For any points $A$ and $B$ in $X$, let $AB:=\|A-B\|$. Suppose that for any points $A$ and $B$ in $X$ and any straight line $\ell\subseteq X$ such that $B\...
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