Questions tagged [similarity]
The similarity tag has no usage guidance.
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An $n \times n$ matrix $A$ is similar to its transpose $A^{\top}$: elementary proof?
A famous result in linear algebra is the following.
An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$.
I know one proof using the Smith Normal Form (SNF). ...
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Some questions on unitarisability of discrete groups
In this post I would like to ask several of questions related to Dixmier problem. I will try to make the post as self-contained as possible.
A discrete group $G$ is unitarisable if for every Hilbert ...
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What is a good algorithm to measure similarity between two dynamic graphs?
I am using graphs to represent structure present in a scene. The vertices represent the objects in the scene and the edges represent the relationship between two nodes(touching, overlapping, none). ...
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How to check two matrices for similitude over $\mathbb{Z}$?
General question. Let $A$ and $B$ be two $n\times n$-matrices over
$\mathbb{Z}$. How do I algorithmically check whether $A$ and $B$ are similar
(i.e., conjugate in the ring $\mathbb{Z}^{n\times n}$)?
...
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Discriminant of elliptic curve (Frey-Hellegouarch), j-invariant and positive definite kernels, similarities?
Consider the Frey-Hellegouarch curve given $a,b$ natural numbers:
$$y^3= x(x-\frac{a}{\gcd(a,b)})(x+\frac{b}{\gcd(a,b)})$$
Then the discriminant is given by $\Delta = \Delta(a,b) = 16 \left(\frac{ab(a+...
user6671
6
votes
2
answers
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similarity transformation into symmetric matrices
I want to determine some structures of matrices that can be transformed into a symmetric matrices using similarity transformation, i.e.,
$B=T^{-1}AT$
where $T$ is the similarity transformation ...
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Are bounded groups of thin operators on Hilbert space similar to groups of unitaries?
QUESTION. Let $G$ be a group of bounded operators on $\ell^2$, satisfying $\sup_{x\in G} \lVert x\rVert <\infty$, whose elements are all of the form "identity+compact" (sometimes called &...
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Looking for techniques of How to measure the Similarity/Dissimilarity between two images?
I would like to compute the similarity/dissimilarity between two images L and R.
I know one way which is : computing the histogram of blocks of each image, and then using Bhattacharyya measure I ...
user74794
5
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0
answers
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Shapes defined by points
Can shapes determined by some number of points?
From an amazing theorem in plane curves geometry we know that vertices of triangles similar to arbitrary triangle $T$ is dense on every closed jordan ...
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Studying finite groups with Euclidean geometry?
Since each finite group $G$ can be considered as a subgroup of the symmetric group, by Cayley's theorem, we might see the elements of $G$ as permutations $\pi$.
Consider for each $\pi \in G$ the set:
...
user6671
3
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1
answer
164
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Similarity via symmetric matrix
Let $K$ be a field extension of $F$. If two $n\times n$ matrices $A,B \in M_n(F)$ are similar via a matrix $P \in GL_n(K)$ (that is, $A=PBP^{-1}$), then there exists a matrix $Q\in GL_n(F)$ such that $...
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1
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Efficient eigen-decomposition of a real matrix with all real eigenvalues
I'm optimising a radar algorithm that results in real matrices which are not symmetric but which are guaranteed to have real eigenvalues. Each matrix is therefore similar to a symmetric matrix. I am ...
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1
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Comparing two distributions based of the ratio of their moments
I am looking for some metric for distribution with support on the interval $[0,1-\epsilon]$, that will be based on the ratio of their moments.
That is, if $X\sim f(x)$, $Y\sim g(y)$, I'm looking for a ...
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Symmetrization of pentadiagonal matrices
Nonsymmetric tridiagonal matrices $T_3$ can easily be symmetrized via a (diagonal) similarity transformation $D=\text{diag}(d_1, \dots, d_n)$ (i.e. see Wikipedia)
$$
J_3=D^{-1} T_3 D \,.
$$
Is there ...
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Sparsest similar matrix
Given a square matrix A (say with complex entries), which is the sparsest matrix which is similar to A?
I guess it has to be its Jordan normal form but I am not sure.
Remarks:
A matrix is sparser ...
2
votes
2
answers
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Appropriate histogram comparison distance measure
I am working with hyperspectral image data in R, so I have subset an image to a region of 5000 pixels, each containing a vector 254 bands in length.
I would like to cluster this data in order to try ...
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0
answers
402
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What does similar eigenvectors and eigenvalues of two matrices really mean? [closed]
Empirically I've noticed that diagonally dominant matrix G and it's diagonal version D (diagonal elements of G on the diagonal and all other elements are set to zero) produce similar eigenvalues and ...
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0
answers
240
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Is there an universal (dis)similarity measure between two structures?
I'm always wondering is there an universal (dis)similarity measure
between two structures (let's say between two undirected simple
graphs)? I mean, not "the measure with universal parameter that we
...
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1
vote
1
answer
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Are there two tetrahedrons with the same volume that share their opposite edge lengths and arent the same or a different chirality of the same? [closed]
I have been coming up with an efficient way to decide if two tetrahedrons are similar. I believe that it is enough for a computer to check for the ordered by length list of pairs of opposite edges on ...
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2
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Similarity measure between 2 bi-partite graph.
Hello there, i need to solve this problem:
I have 2 different bi-partite weighted graph, g1 and g2 and i would like to measure their similarity, g1 and g2 may have different number of vertex and edges ...
- 13
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vote
0
answers
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What is the status of The Halmos Similarity Problem?
What is the general status of "The Halmos Similarity Problem"(HSP) in Operator theory?For What conditions ,HSP has been solved?
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What is a good algorithm to measure similarity between isomorphic graphs with different node labels?
I am using graphs to represent some structured data. In my case, I have a time series of undirected unweighted graphs with the same topology (i.e. isomorphic graphs with same number of nodes and edges,...
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What is the name given to the solution to the equation $cU = Y U Y$ for a given symmetric, positive definite, real-valued matrix $Y$
Overarching question is: What is the name given to the solution to the equation $cU = Y U Y$ for a given symmetric, positive definite, real-valued matrix $Y$? And what procedure is used to solve this ...
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Quantification of the extent of periodicity in a time series using fractal analyses
I need metrics to quantify and compare the extent of periodicity between any two given time series, considering the time series were "almost periodic". By "almost periodic" I mean: if I were to take ...
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187
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Universal Correlation measure — ranking correlations
I have time series data of experimental observations for two related processes. I want to measure correlation for use in further analysis.
Correlation of the series changes over time and across ...
- 111
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2
answers
143
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Simultaneous special orthogonal similarity problem
Given matrices $A,B,C,D\in\Bbb K^{n\times n}$ where $\Bbb K$ is a ring is there an efficient technique to compute set $O$ with $OO'=I$ where $'$ is transpose and $\mathsf{Det}(O)=\pm1$ such that
$$A=...
user76479
0
votes
1
answer
125
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Sizes and shapes of Dedekind cuts
My geometric intuition has failed to tell me that there are different sizes and shapes of Dedekind cuts. I realized it in the course of writing this answer only by doing algebra.
If we define a ...
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How to use node similarity to measure subgraph similarity
For a semantic annotation task I am trying to calculate the semantic similarity between two sets of annotations: S1 and S2. Both sets consist out of multiple nodes from within one graph (in my case an ...
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Fast decay of eigenvector elements
Let A be a set of similar (symmetric) matrices, sharing the same eigenvalues. I understand that their eigenvectors would be different. Let us focus on one eigenvector (e.g. corresponding to the lowest ...
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Comparison of two similarity matrices
English is not my first language, so please excuse any mistakes.
I'm working with two similarity matrices on the same data set: Suppose I have $n$ items, and I calculated the similarity of each item ...
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Eigenvalues of sum of two fuchsian matrices
Dear mathoverflow users,
I am trying to solve a problem concerning eigenvalues and sum of matrices.
In particular: consider the expression
$$
A=\frac{E}{x-x_1}+\frac{F}{x-x_2},
$$
and suppose to know ...
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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 ...
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