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Inverse problem with a rank-1 update

I hope you can help me out with this. I have to find the solution x to an inverse system $$ x=A^{-1}b $$ This inverse problem is basically a least square problem with a rank-1 update. $$ x=[uv^{T}...
user49843's user avatar
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91 views

Complexity of turning a d-degree polynomial to 2-degree polynomial

For a very simple example, $(1+x)^4=x^4+4x^3+6x^2+4x+1$ is a 4 degree polynomial, and I want to change it to a 2-degree polynomial by add more variables, for this example, we can simply let $y=x^2$, ...
ilovecp3's user avatar
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0 answers
251 views

Sparse matrix factorization of a rank deficient matrix by decomposition into linearly independent components

I've got a little conjecture I need to prove for a theoretical result related to causal Bayes net search with latent variables under sparsity constraints. If you're interested in the application ...
Adam's user avatar
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561 views

What are the properties of this linear operator?

Suppose $f(x)$ is a function which satisfies the following condition: $$f(x)=\sum_{k=0}^\infty G(2k)\frac{x^{2k}}{(2k)!}$$ Where the generating function $G(x)$ is a "natural" or "discrete-analytic" ...
Anixx's user avatar
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131 views

The largest size of a boolean subgraph (a hypercube) of a given graph

Let $G(\mathbb{F}_2^n)$ denote the graph that represents the lattice of all subspaces of $\mathbb{F}_2^n$ (also called a Hasse diagram). I am interested in knowing if there exists a large hypercube ...
Spai's user avatar
  • 313
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0 answers
49 views

Forming orthogonal bases in different orders

Let $\alpha_1, \dots, \alpha_n$ be unit vectors in some vector space $V = R^d$. For any permutation $\pi: [n] \rightarrow [n]$, we can form the Gram-Schmidt orthogonal bases $\beta_{\pi,1}, \dots, \...
David Harris's user avatar
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104 views

Linear system with many solutions from a finite set

Basically I am looking for a linear system with many solutions from a finite set. Choose a finite set of rationals $S$ and fix positive integer $k$. Let $A$ be a linear system with $n$ variables $...
joro's user avatar
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320 views

Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial $$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n} $$ After the linear change of ...
zacarias's user avatar
  • 801
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2 answers
153 views

Union of linear inequalities cover whole space?

We have $n$ variables $a_0,a_1,\ldots,a_n$ such that $a_i\geq a_{i+1}$. There are $k$ sets of linear inequality constraints on the $a_i$. I need to check that any choice of $a_i$ satisfies at least ...
Gecko's user avatar
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117 views

"Almost orthogonalizing" matrices using a signature matrix

Suppose $A$ and $B$ are two real symmetric $n \times n$ matrices (If simpler, consider $A$ and $B$ to be 0/1 matrices, say, adjacency matrices of d-regular graphs). Then $||AB||_{op} \leq ||A||_{op}||...
BharatRam's user avatar
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704 views

expected matrix inverse of circulant plus diagonal matrix with chi-square variables

Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$. Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...
john stark's user avatar
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115 views

a very elementary question on the conjugated matrices

Let $A$ and $B$ two matrices in $GL_{n}(K[[\pi]])$, regular semisimple on $GL_{n}(K((\pi)))$, with $K$ an algebraically closed field of characteristic zero . We suppose that they have the same ...
prochet's user avatar
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266 views

Finding the effective maximum number of subspaces in a finite dimensional vector space

Hi mathoverflow community, may be some one may give me a hint on the following problem before I spend much time on brute force search. For $q$ a prime number and $n=6$, let $\mathbb {F}_{q}^{n}$ be ...
R. Simeon's user avatar
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262 views

Lattice basis reductions and finding minimal values

While reading several articles about lattice basis reduction I am left with a few questions. For one, I came across this piece of text Let $\alpha$ and $\beta \in \mathbb{R}$. Also let $X>0$ and $...
Zoe's user avatar
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1 answer
311 views

Subspace generated by positive vectors

Hi everyone, first of all i must admit i'm very familiar with quadratic forms and positive subspaces, so i'm sorry if my question is too trivial. So, here's my problem: Let $L$ be a real vector space ...
Filippo Amaducci's user avatar
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1 answer
775 views

Positive subspaces of quadratic forms

here's my question: Let $V$ be a k-dimensional vector space over $\mathbb{R}$ and $q$ a quadratic form on $V$ of signature $(m,n)$ , $m+n=k$. We have $W\subset V$ a positive (with respect to the ...
Tony Harrison's user avatar
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0 answers
957 views

Diagonal of the inverse of a 6x6 symmetric partitioned matrix

Let $$M = \begin{bmatrix} A & B \\ B & C \end{bmatrix}$$ in which $A$, $B$ and $C$ are $3 \times 3$ matrices being also symmetric. In fact, they are quite similar, just differing on a single ...
Martin's user avatar
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0 answers
52 views

Dense Matrix Estimation

I have a matrix $X \in \mathbb{R}^{m\times n}$ and I want to estimate it with a dense matrix $Y^{m\times n}$ such that $Y$ is still close to $X$ in some distance measure. Is this doable in a ...
Taha's user avatar
  • 137
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0 answers
146 views

Global solution for spectral clustering

I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ...
Fatime's user avatar
  • 13
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1 answer
312 views

Deriving the fundamental equation (with regards to computer vision)

I'm having a hard time understanding how a few equations are being derived. So the fundamental equation is an equation that relates corresponding points in stereo images. Anyway, that's the basic ...
Jared Joke's user avatar
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0 answers
270 views

Solution Existence of a System of Complex Quadratic Equations

Consider $ {x_k}_{k = 1}^K \in \mathbb{C}^{N \times 1}$ a set of $K$ complex vector variables of length $N$. I am interested in finding the existence of a solution of the following quadratic set of ...
mikitov's user avatar
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0 answers
257 views

What is the integer form of a projector into the intersection of the ranges of two integer projection matrices?

Consider two square integer matrices $X$ and $Y$ of the same dimension with the following properties: $X^2=rX$, and $Y^2=sY$ for integers $r$ and $s$. The $\gcd$ of the entries of $X$ is 1 and the $\...
David Clarke's user avatar
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0 answers
155 views

Convexity of a Certain Set of Covariance Matrices

Hello, My question is about a certain set of matrices being convex or not. I'll start with some preliminaries in order to define myself properly. Let $X_1,U,X_2$ be three zero-mean Gaussian random ...
AD1984's user avatar
  • 155
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0 answers
151 views

Ratio of Eigen values and Mutual Independence

Given a matrix $X$. Calculating the Eigen values of $XX^T$ and using the ratio of maximum and minimum eigen values normally gives the condition number of the matrix. If $X$ contains $M$ observations ...
Lepy's user avatar
  • 1
0 votes
1 answer
590 views

Strictly diagonally dominant hermitian matrices eigenvalues sign

Let $A\in \mathcal{M}_{n\times n}(\mathbb{C})$ be a strictly diagonally dominant hermitian matrix. My main goal is to tell how many positive eingenvalues $A$ has in terms of its leading diagonal ...
Linna's user avatar
  • 3
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0 answers
166 views

Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?

Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...
gstar2002's user avatar
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237 views

Geometric Mean of Positive Matrices

Hello all, My question regards the geometric mean (GM) of two positive matrices. The definition of the GM for two positive matrices $(A,B)$ is given by: $M_0(A,B)=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-...
AD1984's user avatar
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0 answers
224 views

When does a real-valued function of a matrix depend only on eigenvalues?

Let $\mathcal{N}$ be the space of all $n \times n$ matrices that are similar to some nonnegative matrix with zero diagonal and let $f: \mathcal{N} \to \mathbb{R}$ be a continuously differentiable ...
Ben Golub's user avatar
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0 answers
244 views

Checking whether this would be bounded

It may be better to post this question here. Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of ...
io0's user avatar
  • 1
0 votes
1 answer
2k views

Finding linearly independent columns of a large sparse rectangular matrix

I have a problem that necessitates solving a large non-negative least-squares problem. My matrix A is large, sparse, highly rectangular (num rows >> num cols) and nearly binary. However, A is not ...
Rob's user avatar
  • 103
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0 answers
324 views

Changing basis on an extension of a free Z-module.

Consider a finite-rank free $Z$-module $Y$. Let $c: Y \times Y \rightarrow Z$ be a $Z$-bilinear form. Assume that $c(y_1, y_2) + c(y_2, y_1)$ is even, for all $y_1, y_2 \in $. Then $c$ "incarnates"...
Marty's user avatar
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0 answers
154 views

linsolve derivative

Consider a vector $\mathbf{g} \in \mathbb{R}^{m}$ and a matrix $\mathbf{A} \equiv \mathbf{A(g)} \in \mathcal{M}_{p\times q} \[\mathbb{R}\]$, a function of $\mathbf{g}$. Furthermore, let $\mathbf{S} \...
user25779's user avatar
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0 answers
204 views

Matrix Mutiplication through Matrix Logarithms and Exponentials

Let $A,B$ be full rank $n \times n$ matrices. If $AB = BA$, then $\exp(\log(A)+\log(B))=AB$. Supposing $A = USL$ and $B = VSL$ where $U,V,S,L$ are integer valued matrices, $det(L)=1$ and $U = LVL^{-1}...
Turbo's user avatar
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0 votes
0 answers
161 views

vector equation

Suppose you have an equation of the form $Hx=Ky$, where $x,y$ are vectors of length $n,m$ respectively ($m>n$) and $H,K$ are matrices of orders $n \times n,n \times m$ respectively. Is there some ...
Felix Goldberg's user avatar
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0 answers
138 views

Approximation of large dimensional vectors by vectors of smaller dimension

sIs there any (efficient) algorithm for the following problem: Let $n = 128$ and $m = 64$ (in the end only $n > m$ matters) and $p_1, \ldots, p_t \in \{ -1,1 \} ^{128}$ be given ($t << 2^{...
tobias's user avatar
  • 397
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0 answers
276 views

Another matrix diagonalization problem

Given the matrices $X$ and $Y$ in $[0,1]^{n\times m}$, for $n > m > 3$, so that $X1_m=1_m$ and $Y1_m=1_m$, where $1_m$ denotes a $m$-length column vector of ones, find a matrix $Q$ in $R^{m\...
silvanmx's user avatar
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0 answers
395 views

The ratio of two strictly increasing functions

Given: \begin{equation} f_1(a)=\sum_{i=1}^{k^*-1} \left(\begin{array}{c} K \\\ i \\ \end{array} \right) \left(-1-\frac{1}{ar}\right)^i \end{equation} \begin{equation} f_2(a)=\sum_{i=1}^{k^*-1} ...
Seyhmus Güngören's user avatar
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0 answers
109 views

Expansion (asymptotic) of scalar function of a square matrix , in terms of determinant of argument?

The title says it all. I have a scalar function (really, a determinant) of a square matrix argument. Can I find an (asymptotic) expansion of the function, in a series in the determinant of the ...
kjetil b halvorsen's user avatar
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0 answers
2k views

In a network with N nodes, what is the general formula for computing the propagation of a set of numbers?

I am creating a circular neural network with N nodes. Each node is connected via a send pathway to every other node, and the connection between two nodes has a weight. Any number sent over the ...
AIGuy's user avatar
  • 1
0 votes
1 answer
655 views

Fuzzy vector similarity

Hi all, I have two multi-dimensional vectors representing documents $\vec{a}$ and $\vec{b}$. Considering cases where there is no overlap between $a$ and $b$ ($a \cap b = \emptyset $), traditional ...
user17528's user avatar
  • 103
0 votes
1 answer
130 views

Maximal length vector under constraints

Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
user16007's user avatar
  • 800
0 votes
0 answers
429 views

[]-infinity algebra and Projective representation

This is a very vague question. We know that some algebra structures can be viewed as modules of some fantastic stuff, call T. Such examples include: Abelian groups are $\mathbb{Z}$-modules, chain ...
Ma Ming's user avatar
  • 1,271
0 votes
0 answers
157 views

Matrices satisfying certain pair-wise constraints

Consider given pairs of variables: $a_{ir1},a_{ir2}\in \mathbb{R}^{m \times m}$ and $a_{jr1},a_{jr2}\in \mathbb{R}^{m \times m}$, where $r \in \{1,2,\cdots,t\}$, consider the constraints: $\sum_{r=1}^...
Turbo's user avatar
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0 votes
0 answers
172 views

Generating Set for $O(V)$ over $\mathbb Z_2$

I am reading a claim that $O(V)$ — the orthogonal group associated with a finite-dimensional vector space $V$ over $\mathbb Z_2$ and a quadratic form $q$, i.e. the group of linear ...
Larry's user avatar
  • 105
0 votes
1 answer
503 views

When are operators extended by linearity bounded?

Greetings. Suppose that $H$ is a separable infinite-dimensional Hilbert space and that $M$ is an infinite dimensional closed subspace of $H$. Suppose that {$v_{n}: n\ge 1$} is an infinite linearly ...
Adam Azzam's user avatar
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0 answers
1k views

Determinant of special generalized Vandermonde matrix

Good evening! I have a generalized Vandermonde matrix of special form: $\left( \begin{array}{ccccc} a_{0,0} & a_{0,1} \cdot x_0 & a_{0,2} \cdot x_0^2 & \ldots & a_{0,m-1} \cdot x_0^{m-...
user avatar
0 votes
0 answers
608 views

Orthogonal Projections in Lie Theory

I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for ...
John Craighead's user avatar
0 votes
1 answer
153 views

Difference of two optimization problem's optimal value

Let we have two following optimization problems: \begin{align} \text{(P1)}\quad \alpha_1 = \max_{x_1,\ldots,x_M} &\quad \sum_{m=1}^{M}\log(1+f_m(x_1,\ldots,x_M))\\ \textrm{s.t.} &\quad \...
Math_Y's user avatar
  • 287
0 votes
1 answer
199 views

Intersection between a line and an n-dimensional parallelotope

Suppose that I have a line in an $n$-dimensional space described by $$ X=A+Bk, \quad \quad X,A,B \in \mathbb{R}^n, k \in \mathbb{R} $$ here $A$ is known and I want to find all the possible vectors $B$ ...
Leonardo's user avatar
0 votes
1 answer
268 views

Nonnegative Matrix

Let $A=E+\sqrt{-1}B$, where $E=diag\{0,1,\cdots,1\}$, $B$ is a real symmetric matrix. Let $A^*$ denote the adjoint matrix of $A$, i.e. $AA^*=\det A\cdot I$. I hope the real part of adjoint matrix ${\...
Yiyan's user avatar
  • 303