All Questions
5,882 questions
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19
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Equality Issue in Deriving Covariance Update for Kalman Filter
I am currently working on deriving the Kalman Gain from the covariance of the updated state and have encountered an equality issue that I am unable to resolve. Below are the derivation steps and the ...
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72
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Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX
I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
- 43
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38
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Sequence of projections that alters a $2^d$ tuple of points to a hyperparallelepiped
Suppose we have a $2^d$ tuple $\{ x_i \}_{i=0}^{2^d-1}$ of points in some $\mathbb{R}^n$. I would like to shift the points of this tuple in some controlled way, so that the final $2^d$ tuple $\{ y_i \}...
- 820
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123
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Realizable singular value spectra of normalized finite frames
$\DeclareMathOperator\tr{tr}$Let $m, n \in \mathbb{N}$, $m \geq n$, and let $\{f_i\}$, $1 \leq i \leq m$, be $m$ unit vectors (wrt. 2-norm) in $\mathbb{R}^n$. Let $A = [f_1 \, \, \, f_2 \, \, \, \...
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36
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Conjugate gradient-like algorithm with multiple search directions
I am solving an $n*n$ system $Ax=b$ in CUDA where $A$ is a sparse matrix. Currently I am solving it using the conjugate gradient algorithm.
I have noticed that $Ax$ where $x$ is $n*1$ has roughly the ...
- 71
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47
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Counting zero-sum subsets of a finite field with a particular form
Let $\mathbb{F}$ be a finite prime field of characteristic different than $2$ and $\beta \in \mathbb{F}$ a generator of the $2$-power order multiplicative subgroup of order $2^k$, so $\beta^{2^{k-1}} =...
- 3,058
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66
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Random elliptical potential lemma
Elliptical Potential Lemma: Let $V_0 \in \mathbb{R}^{d \times d}$ be positive definite and $a_1,a_2,...,a_n \in \mathbb{R}^{d}$ be a sequence of vectors with $||a_t ||_2 \leq L < \infty$ for all $t ...
- 61
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28
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Example of a matrix -HDH that is not PSD (with non-euclidean distances D)
It's widely known that, given a matrix of squared Euclidean distances, $\mathbf{D}_{ij} = \| \mathbf{X}_i - \mathbf{X}_j \|^2$, and the centering matrix $\mathbf{H} = \mathbf{I} - \dfrac{1}{n}11^T$, ...
- 101
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32
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Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
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36
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A class of directed graph, when their minimal polynomial of the adjacency matrix matches the characteristic polynomial
We consider an unweighted directed simple graph, $G$, with a Hamiltonian cycle.
Q. Assume that the adjacency matrix of $G$ is non-singular. Do the characteristic and minimal polynomials of the ...
- 4,058
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68
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Meaning of $\langle M,\,M^{-1}\rangle$
For an $n\times n$ complex matrix $M$, is there a name for the expression $\langle M,\,M^{-1}\rangle$, where the inner product is the Frobenius one, $\langle A,\,B\rangle=\text{tr}(A^*B)$? Is there a ...
- 479
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55
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Johnson-Lindenstrauss type result for matrix factorization
The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
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43
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Given two rectangular matrices and they yield the same results when they are multiplied by their own transposes. What can we say about them?
Suppose we have $MM^T = NN^T$, where $M$ and $N$ are both $n$ by $d$ matrices. Assume that $n$ is (much) larger than $d$, are there anything we could conclude about $M$ and $N$, aside from that $N$ ...
- 9
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64
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When is a symmetric block Toeplitz matrix invertible?
Let
$$
Q =
\begin{bmatrix}
Q_0 & Q_1 & Q_2 & \cdots\\
Q_{-1} & Q_{0} & Q_1 & \cdots\\
Q_{-2} & Q_{-1} & Q_0 & \cdots\\
\vdots & \vdots & \vdots & \ddots
...
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28
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Find a conditional for factorizing the sum of a set of gaussian integer-valued matrices
In my research project, we're exploring the decomposition of Gaussian integer-valued square matrices as a cross-product of other Gaussian integer matrices (GIM) with the same dimension. One of the ...
- 11
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87
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Relation between nullspace and row-equivalence of matrices over $\mathbb{Z}$ and $\frac{\mathbb{Z}}{n \mathbb{Z}}$?
Two matrices $D$ and $E$ over a field have the same nullspace if only if they are row-equivalent. Is the same true if those matrices are over the ring of integers ($\mathbb{Z}$) or integers mod a ...
- 219
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75
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Orbits/affine spaces in GAP
Another GAP-related question.
I need to compute the orbits of a lot (probably, hundreds of thousands) groups acting on $\mathbb{F}_2$-vectors spaces of dimension 23 or 22. The groups range from (...
- 4,959
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43
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Inertia indices in GAP
Not sure that this is the right place, but I could not find a GAP specific forum.
Does anyone know if there is a built-in function in GAP to find the inertia indices of a symmetric matrix, say, over ...
- 4,959
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121
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Representation of anti-commuting matrices in $\mathbb{C}^{2}$
This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem.
The basic question is the following. Let $V$ be a finite-...
- 1,305
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84
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some problem about the discrete of the first derivative operator
I am reading a paper
(Parameter Choice Strategies for Multipenalty Regularization Massimo Fornasier, Valeriya Naumova, and Sergei V. Pereverzyev SIAM Journal on Numerical Analysis 2014 52:4, 1770-1794)...
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67
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Concentration of bilinear forms
This is a bit vague so I'll begin by indicating the motivation. I am looking for ways to [do something interesting or useful] with the self-attention in transformer models. Ultimately the self-...
- 6,962
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184
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Degree 6 Galois extension over $\mathbb{Q} $
Let L be the splitting field of $ x^3- 2$ over $ \mathbb{Q}$. Then $ G=\operatorname{Gal}(L/K) \cong S_3$. Let $\sigma\in G$ such that the fixed field of $ \sigma$ is $\mathbb{Q}(2^{1/3})$. Let $x,y\...
- 923
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79
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Quick calculation of a symmetric product with two indices
Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
- 397
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30
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Application of greedy approach for optimization
I want to maximize an objective given by $$\max_{\{q_n,p_n\}} \sum_{n=0}^\infty (\alpha_1 - \beta_1 n) p_n + (\alpha_2 - \beta_2 n) q_n$$
where $\alpha_1 > \beta_1 >0$ and $\alpha_2 > \beta_2 ...
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108
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Generalization of SVD algorithm
Let $K$ be a field, $A\in K^{n\times m}$ and $\lVert \cdot \rVert$ the Euclidean norm. Consider the problem: Find a $v\in K^m$ such that
\begin{align}
\lVert Av\rVert=\min_{\lVert x\rVert=1}\lVert Ax\...
- 157
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28
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The selection of minimal generating sets in Lie algebra
Suppose $A$ is a Lie algebra on field $F_{p}$ with $[A,A,A]=0$. Denote $\{a_{1},\cdots,a_{d}\}$ is a minimal generating set of $A$.It's possible that $[a_{i},a_{j}]=0$ for some $1\leq i<j\leq d$ ...
- 171
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111
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Totally isotropic space for bilinear pairing over ring
A duplicate of this:
Consider the following well-known inequality: Let $b$
be a non-degenerate symmetric bilinear pairing over a (finite-dimensional) $\mathbb F$-vector space $V$ and $W$
a totally ...
- 223
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310
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Simultaneous triangulation and Jordan normal form of commuting nilpotent matrices
Let $A_1,\ldots,A_r$ be $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers, satisfying $A_i\cdot A_j=A_j\cdot A_i$ for all $i,j$. As the matrices commute, they admit ...
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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 ...
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88
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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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36
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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 ...
- 2,242
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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 ...
- 1,091
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0
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92
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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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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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0
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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
$$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\
...
- 131
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0
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92
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Finding a point that minimizes sum of distances to a given set of lines
Given a set $L$ of size $n$ of lines in $\mathbb{R}^d$, find a point $x \in \mathbb{R}^d$ that minimizes: $$\sum\limits_{l\in L}\min\limits_{y\in l} {\lvert \lvert x-y \rvert\rvert}^2$$
I wrote a 1.5-...
- 19
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0
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87
views
Number of solution to homogeneous linear Diophantine equations
Let $T,M\in\mathbb{N}$ be fixed. Consider a linear Diophantine equation of the form
$a_1 x_1 + a_2 x_2 + … + a_n x_n = 0 $
with $a_i \in [-T,T] \subset \mathbb{Z}$. Is there an asymptotic formula to ...
- 11
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votes
0
answers
83
views
When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?
By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that
$$ C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right] $$
where $e_1,\dots,e_n$ are the standard ...
- 4,058
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votes
0
answers
74
views
Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix
Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
- 4,058
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votes
1
answer
310
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Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$
Given vectors ${\bf v}, {\bf w} \in [0,1]^n$ , where $n \in \mathbb{N} \setminus \{0\}$, and $\alpha > 0$, I would like to find the eigenvalues of the following matrix.
$$\operatorname{diag}({\bf v}...
- 13
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0
answers
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 ...
0
votes
0
answers
102
views
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
0
votes
0
answers
146
views
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
0
votes
0
answers
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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0
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177
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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