All Questions
6,260 questions
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163
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Generalization of polynomial coefficients
I'm dealing with a hard combinatorial problem where for every positive integer value of a variable $n$ I have to calculate a list of numbers, specifically $n^2$, that depend on $n$ and its list index ...
- 47
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votes
0
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88
views
Separating orthogonal vectors in $\mathbb{C}^2$
Is it possible to partition $\mathbb{C}^2$ into two sets $S$ and $S'$ such that, given any two nonzero orthogonal vectors $\mathbf{v}$ and $\mathbf{w}$ of $\mathbb{C}^2$, one of them lies in $S$ and ...
- 679
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19
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Efficient Solution for tridiagonal solving with repeated coefficient lines
I working to speedup calls to LAPACK dgtsv for a specific case, where the the coefficients lines have 2 blocks of repeated coefficients and 3 distinct lines (first, "border" and last)
First ...
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29
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How to synthetize a controller $\dot{u} = F x + G u$ which stabilizes $\dot{x} = Ax + Bu$?
$\textbf{Introduction}$: I study linear control theory. Among strategies, we begin with vector field $Ax + Bu$, $A \in M_{n^2}(\mathbb{R})$, $B \in M_{n \times m}(\mathbb{R})$, and synthesize a ...
- 131
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37
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Telling if matrix is contractive from the spectrum of its Choi-Jamiołkowski isomorphism?
Suppose $T$ is a ${d^2}\times {d^2}$ completely positive matrix, and $M$ is ${d^2}\times {d^2}$ matrix obtained by taking Choi-Jamiołkowski isomorphism of $T$. Is it possible to tell if $T$ is ...
- 2,252
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78
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Let $A, B$ be matrices with elements in $\mathbb{Z}_n$, does $\ker A = \ker B$ imply that they are row equivalent?
Let $A, B$ be matrices with elements in $\mathbb{Z}_n$. If $A x = 0$ and $B x = 0$ have the same set of solutions, where the vectors also have elements in $\mathbb{Z}_n$, does this mean that there is ...
- 219
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196
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Generalized operator norm triangle inequality
Let $O_1, \cdots, O_n$ be Hermitian operators and $c_1, \cdots, c_n$ be complex numbers. If $\| \cdot \|$ denotes the operator norm, does the following inequality hold?
$$\| \sum_{i=1}^N c_i O_i \| \...
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164
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How to prove negativity of a $3\times3$ determinant whose elements involve trigamma, tetragamma, and pentagamma functions?
The classical Euler gamma function can be defined by the integral
\begin{equation*}
\Gamma(z)=\int_0^{\infty}t^{z-1}\operatorname{e}^{-t}\operatorname{d}t, \quad \Re(z)>0.
\end{equation*}
Its ...
- 1,101
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answers
92
views
Norm of matrix product sum
Given matrices $A_{n\times n}, B_{n\times m}, C_{m\times m}$ such that $A^iBC^{N-i}$ is matrix with all zeros except upper right element for all $i$ from $0$ to $N$, what can we say about Frobenius ...
- 159
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68
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Optimal top-k column subset
Let $V$ be a set of vectors over $\mathbb{R}^l$, $l\ge 1$, $\pi_i(V)$ be the permutation of vectors in $V$ such that they are ordered by their $i$th component (descending) in order for $\pi_i(V)(\...
- 101
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answers
43
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Intersection of subspace of cyclical rotations with orthant
In an $N$-dimensional real Euclidian space, let an orthant be specified by a vector
$\underline{x}_0 = \{x_1, x_2, \dots, x_N\}$ where the components $x_k$ are binary in the sense that $x_k = \pm 1$...
- 101
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99
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Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix
Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is
$$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\
...
- 131
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92
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Finding a point that minimizes sum of distances to a given set of lines
Given a set $L$ of size $n$ of lines in $\mathbb{R}^d$, find a point $x \in \mathbb{R}^d$ that minimizes: $$\sum\limits_{l\in L}\min\limits_{y\in l} {\lvert \lvert x-y \rvert\rvert}^2$$
I wrote a 1.5-...
- 19
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87
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Number of solution to homogeneous linear Diophantine equations
Let $T,M\in\mathbb{N}$ be fixed. Consider a linear Diophantine equation of the form
$a_1 x_1 + a_2 x_2 + … + a_n x_n = 0 $
with $a_i \in [-T,T] \subset \mathbb{Z}$. Is there an asymptotic formula to ...
- 11
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83
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When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?
By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that
$$ C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right] $$
where $e_1,\dots,e_n$ are the standard ...
- 4,058
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74
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Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix
Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
- 4,058
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310
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Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$
Given vectors ${\bf v}, {\bf w} \in [0,1]^n$ , where $n \in \mathbb{N} \setminus \{0\}$, and $\alpha > 0$, I would like to find the eigenvalues of the following matrix.
$$\operatorname{diag}({\bf v}...
- 13
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146
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Maximizing the norm of a sum of Hermitian matrices
Consider the following problem:
Problem: Given $n\times n$-Hermitian matrices $A_1,\dots,A_r$, find $e_1,\dots,e_r\in\{-1,1\}$ such that $\|e_1A_1+\dots+e_rA_r\|_\infty$ is maximized. Here the norm is ...
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102
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Construct a vector space whose elements are sets
I would like to construct a vector space whose elements are convex and closed subsets of $\mathbb{R}^n$.
A natural idea is as follows.
For any two sets $S_1, S_2 \subseteq \mathbb{R}^n$, define the ...
- 159
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146
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Linear dynamics in a function space
I posted the same question to Math Stackexchange earlier without much luck, so I am posting here.
I am dealing with a time-dependent model, which can be expressed as a function. $f$ is dependent on ...
- 433
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133
views
On nilpotent singular $\mathbb F_2^{n\times n}$ matrices
Let $M$ be a $0/1$ matrix over $\mathbb F_2^{n\times n}$ with determinant $0$.
The set of such singular matrices form a semigroup.
The set of nilpotent matrices of size $n\times n$ form a semigroup.
...
- 13.9k
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177
views
Given optimality of L1 norm, prove that absolute value of sum of a vector with proper sign is less than 1?
Problem:
Given a domain $\mathcal{D}\subset\mathbb{R}^{l}$, we can find $l$
points $\boldsymbol{v}_{i}\in\mathcal{D}$, $i=1,\cdots,l$. Each
point is a column vector with dimension $l\times1$. They ...
- 1
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205
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Prove that sum of eigenvalues of the inverse of an nxn correlation matrix A is greater than or equal to n
I stuck on this question and here is my thoughts:
So we have a nxn correlation matrix A with eigenvalues: λ_1,λ_2,...,λ_n
1.According to the property of correlation matrix, (λ_1)+(λ_2) + ... + (λ_n) = ...
- 1
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44
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Sufficient conditions to order the solutions to a system of linear equations
A pretty elementary question, but does anyone know of sufficient conditions to order the solutions of a system of linear equations? For example, in the system, \begin{align*}\begin{bmatrix}a_{11}&...
- 1
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172
views
A characterisation of full subgroups of $\mathrm{GL}_n(\mathbf{F}_p)$
Let $p\geq 5$ be a prime and $\mathbf{F}_p$ a finite field of characteristic $p$. A subgroup of ${\rm GL}_n(\mathbf{F}_p)$ is full if it contains ${\rm{SL}}_n(\mathbf{F}_p)$. When $n=2$, we have the ...
- 667
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108
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The eigenstructure of the symmetric tridiagonal matrix whose entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and $$a_{1,2}=\cdots=a_{n-1,n}=1$$
Suppose that $A=(a_{kl})_{k,l=1}^n$ is a symmetric tri-diagonal matix in $M_n(\mathbb{R})$ whose diagonal entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and
$$a_{1,2}=\cdots=a_{n-1,n}=1$$
Any approach to ...
- 4,058
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votes
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answers
126
views
Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same
Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with
$$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$
How can we compute the eigenvectors of $T$?
- 4,058
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126
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On the absolute difference of the Laplacian eigenvalues of an unbalanced signed graph and its underlying graph
Let $\Sigma=(G,\sigma)$ be an unbalanced signed graph with the underlying connected graph $G=(V,E)$ and $\sigma:E\rightarrow \{-1,1\}$, the signing function. Let the Laplacian eigenvalues of $\Sigma$ ...
- 61
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117
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Covering zeroes of quadratic forms by linear forms
Consider a quadratic form
$$Q(x_1,\ldots,x_n)=\sum_{i,j}a_{ij}x_ix_j,$$
where $a_{i,j}\in \mathbb{R}$ and $x_i\in A$ for some $A\subset \mathbb{R}$ such what $|A|=k$.
Question. What is the smallest ...
- 2,288
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145
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Bound on solutions of $Ax \ge b$
Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$.
One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
- 439
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0
answers
112
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What are the properties of square-matrix algebra with this equivalence class?
Consider the set of all square matrices with the following equivalence class:
$\mathbf{A}\sim\mathbf{A}\otimes\mathbf{I}_n$ (or, alternatively, as user @M.G. proposed, $\mathbf{A}\sim\mathbf{I}_n\...
- 10.1k
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273
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Finding the eigenvectors of a submatrix
Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by,
$b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$.
$b_{n+k,l}=...
- 4,058
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votes
1
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71
views
Tensor nuclear norm for a binary 3rd-order tensor
I am interested in the low-rank approximation of a binary(01) third-order tensor. Does anyone know how to define its tensor nuclear norm based on whatever tensor decomposition methods? Could anyone ...
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1
answer
80
views
Can this function be simplified to use only quadratic, linear terms of M with given conditions?
Given the function
$$
E(M) = \sum_{i=1}^N \sum_{a=1}^K \left( M_{ia} \cdot \left\lVert\sum_{i=1}^N M_{ia}\cdot x_i\right\rVert_2^2 \right)
$$
$x$ is a given constant matrix, $x_i$ is a the $n_\text{th}...
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1
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104
views
Do subgradient inequalities hold for matrix convex functions?
Suppose $f$ is a matrix convex function over symmetric, positive semidefinite matrices with spectra in some interval $I$ [1]. That is, for $A,B\succeq 0$ with spectra in $I$, and any $\theta\in[0,1]$,
...
- 253
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149
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Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
- 1
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0
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233
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How to analyse the range of eigenvalues of a symmetric and indefinite matrix?
Let $G$ be a symmetric and indefinite matrix defined as follows
$$ G := S - \begin{pmatrix} I_n & I_n \\ I_n & I_n \end{pmatrix},$$
where $S$ is a symmetric positive definite matrix of size $...
- 1
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votes
1
answer
74
views
$\det(HH’) = 0$ for nonnegative $H$
$H$ is an $n\times m$ matrix with non-negative coefficients and $n < m$. $H'$ is the transpose of $H$.
Are the following statements true?
If $\det(HH’) > 0$, the rows of $H$ define the edges of ...
- 13
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0
answers
78
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Topology of independence set of a vector space
This seems like something that would have a well-known treatment somewhere, but I'm not sure where to look. If we have a vector space $V$ (or maybe even a module), we can consider an abstract ...
- 2,054
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104
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Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$
Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define
$$
u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}).
$$
Question. What are necessary and ...
- 6,853
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0
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174
views
Lipschitz map on positive definite cone of $n$-by-$n$ matrices
A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
- 91
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0
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171
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Multiplying by Loewner-ordered matrices
Suppose $A$ and $B$ are symmetric positive (semi-)definite, and $A<B$ in Loewner order, meaning $B-A$ is positive (semi-)definite. Is it true that, for a symmetric positive-definite $C$, we have $...
- 93
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0
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108
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Solving a nonlinear equation maybe with Lambert W function
Can you please help me solve the following nonlinear equation?
\begin{equation}
\boldsymbol{z} \odot\left(\boldsymbol{\Gamma}^{\top} \boldsymbol{y}\right)=(\beta)^{\frac{1}{m-1}}\left(\frac{m-1}{...
- 11
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0
answers
145
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Given a binary parity check matrix, find a parity check matrix for the same code such that no row has weight greater than $k$
A parity check matrix for a binary linear code is a matrix $H$ (with linearly independent rows) such that the (right) kernel of $H$ is the codespace. Clearly we can replace any row $r_i$ by $r_i + \...
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0
answers
156
views
Optimal solution of complex optimization problem
Let $Q(x)=a(x)e^{jb(x)}$ be a complex function of $x$. We want to approximate this function with $R(x)=\alpha e^{jx\beta}$ such that
\begin{align}
\text{arg}\min_{\alpha,\beta} \int_{-\frac{A}{2}}^{\...
- 287
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
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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 ...
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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
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0
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92
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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