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6 votes
2 answers
623 views

Can this system of equations about Newton's formula have concrete result?

Try to solve this system of equations: $$ S_1=x_1+\dots+x_n=a;\\ S_2=x_1^2+\dots+x_n^2=a;\\ {}\cdots\\ S_n=x_1^n+\dots+x_n^n=a; $$ And find the value of $S_{n+1}=x_1^{n+1}+\dots+x_n^{n+1},a\in\mathbb{...
Er Bu's user avatar
  • 75
1 vote
0 answers
61 views

Discrete-to-continuum convergence of principal Fokker-Planck eigenvalues

I am looking for a reference justifying the following statement. Let $L^n$ be any "reasonably consistent" finite-difference approximation of the Fokker-Planck operator in dimension $d=1$ $$ ...
leo monsaingeon's user avatar
2 votes
1 answer
159 views

Conic hull of a rectangle

I have a simple question that appeared in research: For a rectangle $S :=[a_1,b_1] \times[a_2,b_2] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n$. Let $p_0 = (a_1,a_2,\dots,a_n)$, and define $p_i ...
patchouli's user avatar
  • 275
1 vote
1 answer
179 views

Definition and properties of tangent functional

I am reading Measures Which Agree on Balls by Hoffmann-Jørgensen and I am somewhat confused. Here, $E$ is a Banach space, $S$ is the unit sphere, and $x \in S$. We let $\tau(x, \cdot)$ denote the ...
i like math's user avatar
1 vote
0 answers
82 views

Eigenvalues / vectors of $PQP$, where $P, Q$ are both orthogonal projections

Is there any good way to characterize the eigenvalues / eigenvectors of $PQP$, where $P, Q$ are both orthogonal projection matrices? There are trivial cases when the spaces onto which $P,Q$ project ...
Renat Sergazinov's user avatar
1 vote
1 answer
100 views

Existence of a symmetric matrix satisfying certain irreducible conditions

Let $K$ be a field such that $ \mathrm{char}(K) \neq 2 $. Let $ p(x) $ be an arbitrary irreducible polynomial over $K$ of degree $n$. Using the rational canonical form, we can always construct an $ n ...
Sky's user avatar
  • 923
2 votes
2 answers
235 views

Theoretical/Practical Implications of DFT Eigenvectors

Discrete Fourier transform (DFT) has only four distinct eigenvalues: $±1$ and $±i$. For large matrices , each eigenvalue $λ$ yields a multidimensional eigenspace, allowing linear combinations of ...
ABB's user avatar
  • 4,058
5 votes
0 answers
184 views

What is the fastest algorithm for multiplying one given number with many others?

When multiplying two numbers with each other, which are $n$-bit numbers, there are several algorithms like the one of Karatsuba ($O(n^{\log_2 3})$) and a new one doing it even better (Harvey - Van der ...
tobias's user avatar
  • 749
1 vote
0 answers
112 views

Perron-Frobenius theorem to positive delay differential equations

The Perron-Frobenius theorem is that the largest eigenvalue (in modulus) of a non-negative matrix is real (and simple) and corresponds to a non-negative eigenvector. It is applicable to the positive ...
IscoBerlin's user avatar
3 votes
0 answers
50 views

Stability of indefinitely damped mechanical system with diagonal stiffness

I'm trying to find conditions for the asymptotic stability of the following linear system, \begin{equation} \mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0 \end{equation} given the ...
Shivang Rawat's user avatar
0 votes
1 answer
134 views

Existence of cyclic subspace decompositions for pairs of commuting matrices

Let $\mathbb{K}$ be an arbitrary field (possibly finite). Let $V$ be a finite-dimensional vector space over $\mathbb{K}$, and let $A,B$ be two linear endomorphisms of $V$ which commute. For $v\in V$, ...
Abdelmalek Abdesselam's user avatar
1 vote
0 answers
125 views

Transforming nilpotency into diagonalizability [closed]

We designate the $k$-th standard vector as $e_k$ in $\mathbb{C}^n$. We consider the backward shift operator, denoted as $T: \mathbb{C}^n \to \mathbb{C}^n$, which is defined as follows: $Te_1=0$ and $...
ABB's user avatar
  • 4,058
6 votes
1 answer
1k views

Full expansion of $\det(I+\varepsilon A)$

It is well known that given a $n \times n$ matrix $A$, it holds that $$ \det(I + \varepsilon A)= 1 + \varepsilon \operatorname{tr}(A) + O(\varepsilon ^2).$$ I would need a full representation of $ \...
tommy1996q's user avatar
1 vote
1 answer
138 views

Orthogonal vectors translation using standard vectors

When $n=2m$, let us consider the following vectors $\mathbf{v}_1,\ldots, \mathbf{v}_n$ in $\mathbb{R}^n$ $$\mathbf{v}_q=(v_{1q},\ldots,v_{n,q})$$ $$v_{p,q}=\sin\Big(\frac{pq}{n+1}\pi\Big)$$ It is ...
ABB's user avatar
  • 4,058
2 votes
0 answers
81 views

Degeneracy and the "Linear Degeneracy Testing" problem

The Affine Degeneracy problem is about deciding whether $n$ given points in $\mathbb{R}^d$ (or $\mathbb{Q}^d$) are "in general position". i.e. there is no $d+1$ tuple of points which lies in ...
Tippisum's user avatar
  • 153
2 votes
1 answer
226 views

Inductive Cholesky decomposition to prove that a function is positive definite over the natural numbers?

I am trying to prove that the function: $$k(a,b):=\frac{1}{\operatorname{rad}\left ( \frac{ab}{\gcd(a,b)^2} \right )}$$ is a positive definite function over the natural numbers. What has sometimes ...
mathoverflowUser's user avatar
1 vote
1 answer
118 views

A sine type Chebyshev system

A sequence of real functions $\{\phi_1,\cdots,\phi_n\}$ is called a Chebyshev system on an interval $I\subseteq\mathbb{R}$, if any real linear combination $\sum_{l=1}^n a_l\phi_l$ has at most $n-1$ ...
ABB's user avatar
  • 4,058
3 votes
1 answer
117 views

Show that every stable matrix is similar to a contraction [closed]

Suppose that the square matrix $A$ is stable, i.e., $\rho(A) < 1$. Show that $A$ is similar to a contraction, i.e., show that there exists an invertible matrix $M$ and a matrix $\theta$ such that $...
Quarth's user avatar
  • 31
2 votes
1 answer
135 views

Compact objects in persistence modules and interval decomposition

$\newcommand\Mod{\mathrm{Mod}}\DeclareMathOperator\Fun{Fun}$If $k$ is a field, a persistent $k$-module is a functor $\mathbb{R}\to \Mod_k$ where $\mathbb{R}$ is a poset under the natural ordering of $\...
dicemaster666's user avatar
1 vote
0 answers
72 views

Maximum trace of powers of symmetric $\{0,1\}$-valued matrix with fixed row and column sums

Maximize $\operatorname{tr}(A^k)$ over binary symmetric $n$ by $n$ matrices subject to $$a_{ii}=0, \sum_{j=1}^n a_{ij}=d, \sum_{i=1}^na_{ij}=d,$$ where $d,k$ are fixed positive integers. I am having ...
ComfySofa's user avatar
3 votes
1 answer
273 views

Inflection point calculation for cubic Bézier curve encounters division by zero

I've been working on finding the inflection points of a cubic Bezier curve using the method described in a paper Hain, Venkat, Racherla, and Langan - Fast, Precise Flattening of Cubic Bézier Segment ...
Ziamor's user avatar
  • 133
1 vote
1 answer
270 views

Solve permutation matrix equations of the form: $X^T A X = B_1$ and $X A X^T = B_2$

I have a hard time solving the following two matrix equations for unknown permutation matrix $X \in \mathbb{R}^{n \times n}$: $$X^T A X = B_1$$ $$X A X^T = B_2$$ where, $A$, $B_1$ and $B_2$ are all $n ...
Danish's user avatar
  • 11
36 votes
4 answers
2k views

Determinant of the random matrix $X^2+Y^2$

$\DeclareMathOperator\Prob{Prob}$Let $X,Y\in M_n(\mathbb{R})$ be $2$ random matrices. The entries of $X,Y$ are i.i.d. variables. They follow the standard normal law $N(0,1)$. i) When $n=2,3,4$, one ...
loup blanc's user avatar
  • 3,741
1 vote
0 answers
54 views

Controlling quantity related to Laplacian pseudo-inverse of Erdős–Rényi graph

Consider an $n$-node undirected graph $G = (V, E)$ equipped with weights $W$. Let $L$ be the weighted graph Laplacian matrix, i.e. $L_{ij} = -W_{(i,j)}$ for $(i,j)\in E$ and $L_{ii} = \sum_{j:(i,j)\in ...
yy98's user avatar
  • 11
3 votes
1 answer
210 views

Exponential growth of shortest vector norm for successive lattices corresponding to powers of a matrix

Let $A\in M_{2\times 2}(\mathbb{Z}) $ be a two by two integer matrix such that $0,\pm 1$ are not eigenvalues of $A$ and $\left|\det(A)\right|>1$. I am interested in the growth of the norm shortest ...
an_ordinary_mathematician's user avatar
1 vote
0 answers
36 views

Weighted least squares with matrices as unknowns

Let $M \in \mathbb{R}^{m \times n}$. Let $S \in \mathbb{R}^{m \times N_t}, U \in \mathbb{R}^{n \times N_t}$, with $ N_t \gg m,n$. Moreover, $\epsilon = S - M U$, with $\epsilon$ zero mean white noise ...
baptiste's user avatar
  • 123
2 votes
1 answer
198 views

Eigenvectors and eigenvalues of a symmetric matrix and its entry-wise absolute value

The modulus of matrices is meant componentwise in the following. Let $H$ be a sqaure matrix that satisfies the following assumptions: $H$ is real-valued, symmetric, and positive-definite.. $H$ is ...
keisuke murota's user avatar
0 votes
1 answer
152 views

Name for a monoid on the basis of a vector space?

Is there a name for the structure of a vector space with a monoid defined on its basis? Given a vector space V over a field F, we can choose a basis and define a monoid on it. Now we can use each ...
Spencer Woolfson's user avatar
2 votes
1 answer
40 views

Questions on the "generalized" min. singular value of $A$ given $B$: $\min_{L \in \mathbb{R}^{n \times m}} \{\|BL\|_F: \det(A + BL) = 0\}$

Let $A \in \mathbb{R}^{n \times n}$ be a matrix. Recall $\sigma_{\min}(A)$ is the Frobenius distance between $A$ and the set of singular matrices: $$\sigma_\min(A) = \min_{E \in \mathbb{R}^{n \times n}...
Spencer Kraisler's user avatar
0 votes
1 answer
143 views

From superoperator of unitary to the unitary itself

I have a question about super operators. Let's say I have the super operator of some unitary matrix $u$ called $SU$ where $SU = u^\ast\otimes u$ (here $u^\ast$ is the complex conjugate of $u$). If I ...
Mushahid Khan's user avatar
3 votes
0 answers
112 views

Solving an underdetermined system of linear equations among the rationals, in the neighbourhood of an approximate solution

I have a system of linear equations $Ax=b$. Extremely underdetermined, for concreteness $x \in \mathbb{R}^{17,000}, b \in \mathbb{Z}^{156}$. $A$ is sparse, integer, full rank. I have a very precise ...
Dániel Varga's user avatar
1 vote
1 answer
217 views

Perturbation of matrices

Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$. Question. Does there exist a Lebesgue measurable ...
Ali's user avatar
  • 4,135
1 vote
0 answers
91 views

Positive semidefinite maximum principle

Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Let $\mu$ be a Borel probability measure on $M_n(K)$ supported on a compact set $C$ of positive semidefinite matrices with $\mathbf{0}\not\in C$. ...
Joseph Van Name's user avatar
13 votes
2 answers
730 views

Concrete representation of coend in linear algebra

$\require{AMScd}$Teaching coend calculus to a PhD student led me to this "elementary" computation that I would like to perform explicitly. Consider the functor $F : (\mathbb N,\le)^\text{op}\...
fosco's user avatar
  • 13.6k
1 vote
0 answers
27 views

Spectrum of the convolution of the Maxwell collision kernel with a distribution

Given the Maxwell collision kernel $A(z) = |z|^2I_d - z \otimes z$, where $I$ denotes the $d\times d$ identity matrix and $z\otimes z = zz^T$ is the outer product, it is easy to see that $A(z)$ has ...
Vasily Ilin's user avatar
12 votes
1 answer
902 views

Positive 4-form

Denote by $W$ the space of all symmetric bilinear forms on $\mathbb{R}^n$. Let $Q$ be a quadratic form on $W$. Suppose that $Q(b)\geqslant 0$ for any $b\in W$ such that $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
Anton Petrunin's user avatar
2 votes
1 answer
217 views

How to do LU factorization efficiently based on the factorized result added with a low-rank matrix?

Suppose a square $n\times n$, dense matrix $A^{\text{old}}$ has been factorized into $L^{\text{old}}$ and $U^{\text{old}}$ components by performing a LU decomposition $A^{\text{old}} = L^{\text{old}}U^...
Alex Joe's user avatar
2 votes
0 answers
107 views

Two definitions for transverse $(p,p)$ form

Let $V$ be a complex vector space of dimension $n$ and let $V^*$ be its dual. Fix any integer $1\leq p\leq {n-1}.$ A $(p,p)$-form $\alpha\in\bigwedge^{p,p}V^*$ is said to be strictly weakly positive ...
Invariance's user avatar
2 votes
1 answer
200 views

Upper bound for the rank of a Gram-type matrix

Let $V = (v_{1},...,v_{N})$ and $W = (w_{1},...,w_{N})$ be 2 sets, each containing $N$ vectors from $\mathbb{R}^{n}$; i.e, $v_{j}, w_{j} \in \mathbb{R}^{n}$ for all $1 \leq j \leq N$. Assume that $N$ ...
Tomer Milo's user avatar
2 votes
1 answer
207 views

Orthogonal complements in exterior powers

I previously asked this on Mathematics Stack Exchange, to no result: Consider the standard induced inner product structure on $\wedge^k\mathbb{R}^d$ given by defining $$\langle u_1\wedge \cdots \wedge ...
Ian Morris's user avatar
  • 6,206
2 votes
1 answer
298 views

Is there a combinatorial interpretation for the change of basis matrix in the Frobenius normal form representation?

Let $G$ be a graph on $n$ vertices. Let $A$ be the adjacency matrix of $G$ (i.e., rows and columns of $A$ are indexed by vertices of $G$, and the $(v,w)$ entry of $A$ is $1$ if $(v,w)$ is an edge in $...
Naysh's user avatar
  • 557
1 vote
0 answers
79 views

A general theory of pairings

Bilinear forms and bilinear maps for vector spaces over a field are standard material for an introductory course in linear algebra. There are also text books for bilinear forms and related quadratic ...
Thomas Preu's user avatar
5 votes
1 answer
210 views

Relation between row space and column space resp. null space and left null space over general rings

Let $R$ be a ring and $M\in\text{Mat}(R,m\times n)$ a matrix for $m,n\in\mathbb{N}$. What results are known about the relation between column space (cs, image) and row space (rs), resp. null space (...
Thomas Preu's user avatar
3 votes
1 answer
198 views

Do radially bounded sets form a bornology?

We call a subset $A$ in a real vector space $E$ radially bounded if it intersects every ray emanating from $0$ via a bounded set. It is easy to see that radially bounded sets in $E$ form a bornology, ...
erz's user avatar
  • 5,529
1 vote
0 answers
134 views

Matrix valued word embeddings for natural language processing

In natural language processing, an area of machine learning, one would like to represent words as objects that can easily be understood and manipulated using machine learning. A word embedding is a ...
Joseph Van Name's user avatar
15 votes
2 answers
1k views

Positive quadratic polynomial

Let $S$ be solutions of a system of quadratic polynomials on $\mathbb{R}^n$. Suppose $q$ is another quadratic polynomial such that $q|_S\geqslant 0$. Is it possible to find a polynomial $\tilde q$ ...
Anton Petrunin's user avatar
0 votes
0 answers
164 views

How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?

The classical Euler gamma function can be defined by the integral \begin{equation*} \Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0. \end{equation*} Its ...
qifeng618's user avatar
  • 1,091
2 votes
0 answers
200 views

On the inequality-integer system [closed]

I need to prove this inequality, but I do not have a good background in algebra, if you can guide me: We have: $$ p_1 + 2p_2 + \ldots +kp_k < q_1 + 2q_2 + \ldots +kq_k+(k+1)q_{k+1}+\ldots+tq_t $$ ...
BADJARA Mohamed el Amine's user avatar
0 votes
1 answer
171 views

Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$

Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group $G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
ghc1997's user avatar
  • 823
3 votes
0 answers
79 views

When can the eigenvalues be constrained to the right half plane as a result of column permutations?

Suppose we have a real square matrix. Under what conditions is it possible to permute the columns of the matrix such that all eigenvalues of the resulting matrix have nonnegative real part?
Max Aifer's user avatar

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