All Questions
6,287 questions
0
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34
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It is possible to limit a set of curves in the sense $f(x,y) \leq C f(x_0,y)$?
Suppose you have a continuous function $f:[a,b]\times (-\infty, \infty) \rightarrow \mathbb{R}$. I'm trying to understand if it's possible to conclude that due to the compactness of the interval $[a,b]...
- 677
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 $...
- 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 $ \...
- 697
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 ...
- 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 ...
- 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 ...
- 3,064
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 $n-1$ ...
- 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 $...
- 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 $\...
- 213
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 ...
- 11
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 ...
- 133
3
votes
1
answer
128
views
Weaker version of the lemma of K.L. Chung
Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds:
$$...
- 132
1
vote
1
answer
270
views
Solve permutation matrix equations of the form: $X^T A X = B_1$ and $X A X^T = B_2$
I have a hard time solving the following two matrix equations for unknown permutation matrix $X \in \mathbb{R}^{n \times n}$:
$$X^T A X = B_1$$
$$X A X^T = B_2$$
where, $A$, $B_1$ and $B_2$ are all $n ...
- 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 ...
- 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 ...
- 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 ...
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 ...
- 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 ...
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 ...
- 481
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}...
- 518
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 ...
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 ...
- 31
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 ...
- 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 $\mathbf{0}\not\in C$. ...
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}\...
- 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 ...
- 43
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 $b(X,Y)=\ell(X)\cdot\ell(Y)$ ...
- 45k
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^...
- 21
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 ...
- 295
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 $N$ ...
- 93
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 ...
- 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 $...
- 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 ...
- 231
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 (...
- 231
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, ...
- 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 ...
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$ ...
- 45k
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 ...
- 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
$$
...
0
votes
1
answer
171
views
Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
- 823
3
votes
0
answers
79
views
When can the eigenvalues be constrained to the right half plane as a result of column permutations?
Suppose we have a real square matrix. Under what conditions is it possible to permute the columns of the matrix such that all eigenvalues of the resulting matrix have nonnegative real part?
- 31
7
votes
1
answer
633
views
Given a rational matrix $Q$, can we generate $\langle Q^{i}(v)\mid i\in\mathbb Z,v\in\mathbb Z^{2}\rangle$ using only non-negative powers of a matrix?
I have copied this question from StackExchange, thank you to those who helped me to improve the question. (apology if you have seen this question already)
Let $Q $ be a matrix in $ \operatorname{GL}(...
- 823
0
votes
1
answer
298
views
Is there a reference on the space of Lipschitz continuous functions?
I have hard a time finding the specific properties I'm looking for, I'm wondering if there is literature which proves (or disproves) that the space of all Lipschitz continuous functions of some ...
0
votes
2
answers
131
views
Reshaping data vector into a matrix for deconvolution using a circulant matrix
Suppose we have a circulant matrix S made from pseudorandom binary sequence of length $N$ consisting of $0$'s or/and $1$'s. $1$ means that we can inject something for chemical analysis and $0$ means ...
- 879
2
votes
1
answer
200
views
An inequality related to matrix trace
$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ where $A \in \mathbb{R}^{m\times n}$ is a real rectangular matrix, $U \in \mathbb{R}^{n \times n}$ is a orthonormal matrix and $\Sigma$ is a ...
- 21
4
votes
0
answers
219
views
Map $\operatorname{Sym}^{mp}(V^*) \longrightarrow K^{q}$ defined by $q$ points in $\operatorname{Sym}^p(V)$
EDIT : I have edited the question and made it more specific with respect to the kind of answer I expect.
Let $V$ be a finite dimensional $K$-vector space and let $x_1, \dotsc, x_q \in V$ be $q$ points,...
- 7,300
2
votes
1
answer
247
views
Linear system with matrix as a variable
I have the following two linear systems:
$$\begin{bmatrix} u_{11} & u_{12} \end{bmatrix} A = 0$$
$$\begin{bmatrix} u_{21} & u_{22} \end{bmatrix} B = 0$$
Both $A,B$ are $2 \times 2$ matrices ...
- 129
0
votes
0
answers
92
views
Norm of matrix product sum
Given matrices $A_{n\times n}, B_{n\times m}, C_{m\times m}$ such that $A^iBC^{N-i}$ is matrix with all zeros except upper right element for all $i$ from $0$ to $N$, what can we say about Frobenius ...
- 159
0
votes
0
answers
68
views
Optimal top-k column subset
Let $V$ be a set of vectors over $\mathbb{R}^l$, $l\ge 1$, $\pi_i(V)$ be the permutation of vectors in $V$ such that they are ordered by their $i$th component (descending) in order for $\pi_i(V)(\...
- 101
0
votes
1
answer
67
views
Multivariate random variable problem [closed]
I'm stuck on this problem:
Let a random vector $X$ be given in $\mathbb{R^{10}}$ with the standard scalar product. It is known that $\mathbb{E}[XX^T] = 5I_{10}$, $I_{10}$ – identity matrix of order 10....
- 69