All Questions
1,923 questions with no upvoted or accepted answers
0
votes
0
answers
114
views
Degeneracies in linear combination of tensor product of Pauli matrices
Let $P_i \in \{I,X,Y,Z\}^{\otimes n} $, that is $P_i = \bigotimes_{i =1 }^n \sigma_i$ with $\sigma_i \in \{I,X,Y,Z\}$, where
$$
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \hspace{1cm} X =...
- 131
0
votes
0
answers
37
views
Linear system with +-1 coefficients and three variables for each equation
I have a linear system $LS$, where each equation contains three variables and the coefficient of each variable is $\pm 1$. For example, I have $x_{a}-x_{b}+x_{c}=p$ ($p$ is the known term).
Suppose ...
0
votes
0
answers
100
views
Invertible matrices with bounded nonnegative coefficients
I am teaching a class in linear algebra and I asked myself the following question: what is the chance to get an invertible matrix if I write a random one? My impulsive answer is "very likely"...
- 2,152
0
votes
0
answers
92
views
Positive definite matrix and Hörmander theory
Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$
Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set
$$
\varphi_{\...
- 101
0
votes
0
answers
56
views
Determining the total number of nonzero expansion terms in a (0,1)-matrix
Let $A=(a_{ij})_{n\times n}$ be a $(0,1)$-matrix such that it contains equal number of $1$s in each row and column. Is there any general method to count the total number of the nonzero terms $\prod_{i=...
- 133
0
votes
0
answers
151
views
CRT for linear forms
Suppose $A,B,C,A',B',C'$ are random distinct primes in $[T,2T]$ and $u,v$ are integers in $[T,2T]$.
Suppose we know:
$$Au+Bv\equiv r\bmod C$$
$$A'u+B'v\equiv r'\bmod C'$$
can we identify $u,v$ in ...
- 13.9k
0
votes
0
answers
180
views
Adding the AWGN to the data makes its covariance matrix always positive definite?
I'm working on a numerical method that estimates direction-of-arrivals in antenna arrays.
I realized that every time I add the AWGN (Additive white Gaussian noise) to a data (which is a matrix), its (...
0
votes
0
answers
92
views
Classification of elements $GL(d, \mathbb{R})$
Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here.
Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
- 1,043
0
votes
0
answers
57
views
Determinant of barycenter of a hyperbolic-matrix
Let $A \in \operatorname{GL}(d, \mathbb{R})$ be a hyperbolic matrix. I want to show that $$\det((1-\alpha)A+\alpha\operatorname{Id})\geq 1,$$
where $0<\alpha<1$.
Attempt:
In $\operatorname{SL}(2,...
- 1,043
0
votes
0
answers
67
views
Integration of matrix form of Vasicek variance (Python/Matlab)
$X_t$ is a vector and follows the following Vasicek process.
$$
dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\
$$
What is the variance of $X_t$?
In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...
- 1
0
votes
1
answer
397
views
What is the best way to choose initial basis when applying simplex method to an equality form of LP?
Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
- 1
0
votes
0
answers
133
views
A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation"
Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126."
On page 125, at the end of the proof of Theorem 4.3, I abstract ...
0
votes
0
answers
105
views
Unitarily equivalent matrices that are also unitarily equivalent on orthogonal subspaces
Consider two positive semidefinite matrices $A$ and $B$ on $\mathbb C^d$.
Let $\{P_i\}_{i=1}^m$ be a complete family of $m$ orthogonal projectors on $\mathbb C^d$ (i.e., $P_i^*=P_i, P_iP_j=\delta_{ij}...
- 31
0
votes
0
answers
124
views
The best unitary matrices that approximate a matrix product
Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
- 287
0
votes
0
answers
46
views
What kind of bounds for $\mathrm{Re}(\lambda(A))$ when $\lambda_{\mathrm{max}}(A + A^t) < 0$?
What can be said about the real parts of eigenvalues of $A \in \mathbb R^{n\times n}$ when $\lambda_{\max}(A + A^t) < 0$?
I think the real parts of eigenvalues of $A$ will be negative, but I can't ...
- 1
0
votes
0
answers
112
views
How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?
Let $ A = \begin{bmatrix}
a & 1 \\ 0 & a
\end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $.
Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
- 101
0
votes
0
answers
253
views
Is there a relationship between infinity norm (or any other norms) of a vector and the trace of its covariance matrix?
I wish to know if there is a known relationship between the infinity norm (or any other norms) of a vector and the trace of its covariance matrix?
I have found a paper that used the following ...
0
votes
0
answers
112
views
Existence of a subspace of having no isotropic 2-plane
Let $V$ be a vector space of dimension $n$ over the field $\mathbb {Q} $. A subspace $W$ is isotropic for a skew-bilinear form $\alpha$ on $V$ if $\alpha(x,y) = 0$ for all $x,y \in W$.
More ...
- 923
0
votes
0
answers
198
views
eigenvalues of the product of a unitary with a diagonal
In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
- 4,058
0
votes
0
answers
253
views
Determinant of chain complexes
Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of ...
user30211
0
votes
0
answers
44
views
Polynomial representation with shared root
Let $R$ be a polynomial quotient ring of the form $F_3[x_1,x_2,...,x_n]/\langle \{x_i^3-x_i\}_{1\le i\le n} \rangle$ and $\{f_i\}_{1\le i \le m}$ be elements of $R$, where $m > n$. We know that the ...
0
votes
0
answers
150
views
Sequence in matrices converging to Identity
Let $M_{k}(\mathbb{C})$ be the set of $k \times k$ complex matrices. I am trying to find a sequence of polynomials $P_{n}: M_{k}(\mathbb{C}) \to M_{k}(\mathbb{C})$ (or continuous functions $f_{n}$) ...
- 31
0
votes
0
answers
96
views
Books on limiting properties of matrices with growing size
This question has been posted on Math-Se previously.
I am studying asymptotic properties of the Projection Matrix
$$
H_n=X'(X'X)^{-1}X
$$
By the Gerschgorin disc theorem, the bounds on the ...
- 101
0
votes
0
answers
41
views
Combining coefficients in a sum of inequalities
In an induction proof of a lemma I would like to prove the following statement.
Let $U$ be a non-empty finite set and let $X_{i,j}$ for all $i\not=j \in U$ be real numbers. Assume for each $j\in U$ we ...
0
votes
0
answers
257
views
What's the definition of Euclidean density?
In page 19 of the article "Equivariant cohomology with generalized coefficients" the authors say:
Let $V $ be an n-dimensional real vector space, let $v'$ be a non-zero element in $ \wedge^...
- 139
0
votes
0
answers
54
views
Rank decomposition of matrices over $\mathbb F_2$
Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$?
If $...
- 13.9k
0
votes
0
answers
135
views
Is there an efficient algorithm to project a vector onto the eigenbasis of a symmetric matrix?
Let $H$ be a symmetric matrix over $\mathbb R^n$. Given some vector $u$, I would like to express $u$ in the eigenbasis for $H$. Can this be done efficiently, perhaps using some kind of iterative ...
- 623
0
votes
0
answers
105
views
Any open subgroup of $GL_n(K)$ contains $U_n(K)$
My setting is the $p$-adic field. $K$ (finite extension of $\Bbb Q_p$.) Proposition 2.1.18 seems to claim that
Any open subgroup of $GL_n(K)$. contains $U_n(K)$ the unipotent upper triangular ...
- 661
0
votes
0
answers
104
views
Efficient Algorithm to Find Subset of Vectors Over $\mathbb{F}_q$ Living in Low Dimensional Subspace
Let $q$ be a fixed prime, $P, Q$ be polynomials with $\mathrm{deg}(Q) < \mathrm{deg}(P)$ and $h = O(\log n)$.
Let $S$ be a subset of $\mathbb{F}_q^n$ of size $P(n)$ such that there exists a subset ...
- 328
0
votes
0
answers
183
views
Eigenbases without the Axiom of Choice
I understand that in ZF set theory without the Axiom of Choice (AC), it is consistent to have models in which there exist vector spaces over some (unspecified) field $k$ without a basis.
So in ...
- 4,605
0
votes
0
answers
137
views
Any technique for linearization, or linear approximation?
Consider the following Matrix constraint:
$$
\begin{bmatrix} -U+\psi\Sigma_b^{-1} & V \\ V^T & -V^TU^{-1}V+\tau_2 -\psi \end{bmatrix} \leq 0
$$
where $\Sigma_b$ is a known positive definite ...
0
votes
0
answers
228
views
Decomposition of symmetric block matrix
I came across this question and got really interested about it. There, the OP asks whether is possible to decompose a $2n \times 2n$ block matrix:
$$ \begin{pmatrix}
X & I \\
I & Y
\end{...
0
votes
0
answers
46
views
Lipschitz solutions to linear complementarity problems (LCP)
Let $M\in\mathbb{R}^{n\times n}$.
For $q\in\mathbb{R}^n$, define the set:
$$S_M(q)=\{y\in\mathbb{R}^n|y\ge 0,q+My\ge 0, y^\top (q+My)=0\}.$$
This is the set of solutions to the LCP $(q,M)$.
We say $...
- 183
0
votes
0
answers
113
views
Finding which members of a family of (possibly infinite-dimensional) matrices have trivial null space
Background
I have a set $S$ (that is possibly infinite) and a correspondence between functions $c:S^3\to\mathbb{R}$ (I will write $c(i,j,k)$ as $c_{ijk}$) and matrices $M$ with rows indexed by $(i,j,k)...
- 509
0
votes
0
answers
194
views
Rewriting Kronecker product
im considering a pole placement problem in control theory and my controler has a specific form:
$$R=I_n\otimes q$$
where $I_n$ is the identitiy matrix of size $n$ and $q\in\Re^k$ is a vector of the ...
- 1
0
votes
0
answers
331
views
Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere
Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
- 6,853
0
votes
0
answers
94
views
Large subgroups of infinite-dimensional vector spaces
Let $V$ be an infinite-dimensional vector space over $\mathbb{Q}$.
Consider a proper subgroup $W$ of $V, +$ with the following property: each vector line $L$ (which we see as a subgroup of $V, +$) has ...
- 4,605
0
votes
0
answers
88
views
Fast decay of eigenvector elements
Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest ...
- 47
0
votes
0
answers
69
views
Formulas involving traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices
While working on the Atiyah problem on configurations of points, I came across formulas involving products of traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices.
To ...
- 5,215
0
votes
0
answers
251
views
What is the computational complexity of solving a highly underdetermined system?
Let $F$ be a finite field with $q$ elements. Consider an underdetermined system of linear equations with $m$ equations and $n$ variables where $n\gg m$. What is the complexity of solving such a highly ...
- 131
0
votes
0
answers
129
views
Symmetry of points on unit sphere determined by relation between triples of points
Suppose we have $n$ points on the 3D unit sphere,
$X = (\pmb{r}_{1}, \pmb{r}_{2}, ..., \pmb{r}_{n})$.
I am interested in knowing to what extent the rotational symmetry of $X$ is determined by the ...
0
votes
0
answers
78
views
The Hilbert matrix becomes degenerate after slight modification
Let $H_{n+1}$ denote the $(n+1)\times (n+1)$ Hilbert matrix, i.e., the $(i, j)$-entry of $H_{n+1}$ is $(i+j-1)^{-1}$ with $1 \leq i, j \leq n+1$. Let $A$ denote an $(n+1)\times (n+1)$ matrix whose ...
- 1
0
votes
0
answers
146
views
Square root of a circulant matrix block
I'm trying to show the following:
Given the following $n\times n$ symmetric circulant matrices
$$A^*=\begin{pmatrix}
1 & -\mu_a & 0 & ...&0&-\mu_a \\
-\mu_a & 1 & -\mu_a &...
0
votes
0
answers
57
views
Numerically finding matrix approximation by lower-dimensional "pseudo-similar" matrix
Consider an $N\times N$ (real or complex) matrix $A$, and some $n<N$. Is there a good numerical algorithm that finds the set consisting of an $n\times n$ matrix $B$, an $n\times N$ matrix $I$, and ...
- 3,001
0
votes
0
answers
299
views
Question on rank of matrices over $\mathbb F_2$
$A$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $k\leq n-1$.
$B$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $n$.
$T$ is a square matrix in $\mathbb F_2^{n\times n}$ of rank $1$...
- 13.9k
0
votes
0
answers
90
views
Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?
Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\tag{1}S^{-1}\cdot A\cdot S = \...
- 463
0
votes
0
answers
195
views
Proof negative-definiteness of a nonsymmetric and rank-deficient matrix
Consider the vectors $\mathbf{a} \in \mathbb{R}^N$ and $\mathbf{b} \in \mathbb{R}^N$ with $N>1$ and $\mathbf{a} \neq \mathbf{b}$.
The product $\mathbf{C}=\mathbf{a} \mathbf{b}^T \in \mathbb{R}^{N \...
- 1
0
votes
0
answers
425
views
Linear independence of vectors in Graph Theory
I have poste this question on StackExchange but there were no takers - would I be luckier on this site?
Most of this is well known, so let me just restate the corresponding Math:
Given a connected, ...
- 419
0
votes
0
answers
141
views
Parseval's equivalent of Norm that includes a Projection matrix
I need to optimize the norm, ${\bf x}^H {\bf P}_{\bf B} {\bf x} $, where, ${\bf P}_{\bf B} = {\bf B}^H({\bf B} {\bf B}^H)^{-1} {\bf B}$, ${\bf B}$ is a known $M \times N$ matrix, with $M < N$ and $...
- 1
0
votes
0
answers
25
views
Concentrate singular values by diagonal similarity
Given non-singular real matrix $A$ of size $n \times n$. I want to find a non-singular diagonal matrix $D$ such that
$$
B = D A D^{-1}
$$
has singular values $\sigma(B) = [1, \dots, 1, \lvert \det(A) ...
- 909