Questions tagged [similarity]

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153 views

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: ...
4
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0answers
259 views

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+...
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0answers
112 views

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 ...
3
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0answers
76 views

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 ...
3
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1answer
107 views

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 ...
3
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0answers
99 views

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 ...
1
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1answer
98 views

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 ...
3
votes
1answer
148 views

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 $...
0
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2answers
110 views

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=...
0
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0answers
60 views

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 ...
0
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1answer
113 views

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 ...
2
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0answers
313 views

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 ...
1
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0answers
154 views

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 ...
5
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3answers
1k views

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 ...
2
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0answers
223 views

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 ...
10
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4answers
12k views

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). ...
1
vote
2answers
959 views

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 ...
6
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2answers
3k views

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 ...
23
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0answers
5k views

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). ...
1
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2answers
890 views

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 ...
1
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0answers
181 views

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 ...
0
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1answer
1k views

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 ...
0
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1answer
595 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 ...
14
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0answers
589 views

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 ...