A generating function for averages over the orthogonal group

1955 ◽  
Vol 229 (1178) ◽  
pp. 367-375 ◽  
Keyword(s):  
Invariant Measure ◽  
Generating Function ◽  
Orthogonal Group ◽  
Wishart Distribution ◽  
Orthogonal Matrix ◽  
Orthogonal Matrices

The integral, ∫ H exp ( tr H ′ X ) d V ( H ) , over the group H of orthogonal matrices with respect to the invariant measure V ( H ), is obtained. X is an n × n matrix, and H an n × n orthogonal matrix. Various applications of it are discussed, namely, to the non-central Wishart distribution and as a generating function for the integrals and averages of polynomials over the orthogonal group.

scholarly journals ROTATION MATRIX SAMPLING SCHEME FOR MULTIDIMENSIONAL PROBABILITY DISTRIBUTION TRANSFER

2016 ◽  
Vol III-7 ◽  
pp. 103-110
Author(s):  
P. Srestasathiern ◽  
S. Lawawirojwong ◽  
R. Suwantong ◽  
P Phuthong
Keyword(s):  
Probability Distribution ◽  
Orthogonal Matrix ◽  
Rotation Matrix ◽  
Sampling Scheme ◽  
Uniform Rotation ◽  
Householder Transformation ◽  
Orthogonal Matrices ◽  
Matrix Sampling ◽  
Paper Address ◽  
Multidimensional Probability

This paper address the problem of rotation matrix sampling used for multidimensional probability distribution transfer. The distribution transfer has many applications in remote sensing and image processing such as color adjustment for image mosaicing, image classification, and change detection. The sampling begins with generating a set of random orthogonal matrix samples by Householder transformation technique. The advantage of using the Householder transformation for generating the set of orthogonal matrices is the uniform distribution of the orthogonal matrix samples. The obtained orthogonal matrices are then converted to proper rotation matrices. The performance of using the proposed rotation matrix sampling scheme was tested against the uniform rotation angle sampling. The applications of the proposed method were also demonstrated using two applications i.e., image to image probability distribution transfer and data Gaussianization.

scholarly journals A note on p-adic Carlitz's q-Bernoulli numbers

2000 ◽  
Vol 62 (2) ◽  
pp. 227-234 ◽  
Author(s):  
Taekyun Kim ◽  
Seog-Hoon Rim
Keyword(s):  
Invariant Measure ◽  
Generating Function ◽  
Bernoulli Number ◽  
Bernoulli Numbers ◽  
Adic Case

In a recent paper I have shown that Carlitz's q-Bernoulli number can be represented as an integral by the q-analogue μq of the ordinary p-adic invariant measure. In the p-adic case, J. Satoh could not determine the generating function of q-Bernoulli numbers. In this paper, we give the generating function of q-Bernoulli numbers in the p-adic case.

The conjugate classes of the cubic surface group in an orthogonal representation

1955 ◽  
Vol 233 (1192) ◽  
pp. 126-146 ◽  
Keyword(s):  
Orthogonal Group ◽  
Surface Group ◽  
The Other ◽  
Identity Matrix ◽  
Orthogonal Matrices ◽  
Linear Spaces ◽  
Cubic Surface ◽  
Orthogonal Representation ◽  
Conjugate Class ◽  
Group A

The cubic surface group, of order 51840, has a representation by orthogonal matrices, of 5 rows and determinant + 1, over GF (3). It can be partitioned into conjugate classes on geometrical grounds because each matrix has two skew linear spaces, S + of even and S - of odd dimension, of latent points; the matrices fall into categories A, B, C according as the join of S + and S - has dimension 4, 2, 0. Subdivisions of A, B, C rest on the relation of S + and S - to the invariant quadric of the orthogonal group. A accounts for the identity matrix and the 4 types of involutions. B falls into two parts; one of 4 classes, discussed in §§5 to 8, the other of 9 classes, discussed in §§9 to 14. §§ 15 and 16 mention criteria for checking the number of operations in a conjugate class. Those classes in category C fall into 3 subcategories of 3, 2, 2 classes and are described in §§ 18 to 25.

Orthogonal Matrices in Four-Space

1949 ◽  
Vol 1 (1) ◽  
pp. 69-72
Author(s):  
C. C. MacDuffee
Keyword(s):  
Orthogonal Transformation ◽  
Orthogonal Matrix ◽  
Orthogonal Matrices ◽  
Image Position ◽  
Proper Orthogonal ◽  
The Relationship

Every proper orthogonal matrix A can be writtenwhere Q is a skew matrix [6], and conversely every such matrix A is orthogonal. It is also known that every proper orthogonal transformation in real Euclidean four-space may be characterized in term of quaternions [1, 3] by the equationdetermines with the origin a vector having the coordinates (XQ, XI, x2, x3). The relationship between these two representations was clearly shown by Murnaghan [5].

scholarly journals Sign patterns of J-orthogonal matrices

2017 ◽  
Vol 5 (1) ◽  
pp. 225-241
Author(s):  
Frank J. Hall ◽  
Zhongshan Li ◽  
Caroline T. Parnass ◽  
Miroslav Rozložník
Keyword(s):  
Orthogonal Matrix ◽  
Orthogonal Matrices ◽  
Sign Patterns ◽  
Exchange Operator ◽  
Czechoslovak Math ◽  
Siam Review ◽  
G Matrices ◽  
Block Diagonal

Abstract This paper builds upon the results in the article “G-matrices, J-orthogonal matrices, and their sign patterns", Czechoslovak Math. J. 66 (2016), 653-670, by Hall and Rozloznik. A number of further general results on the sign patterns of the J-orthogonal matrices are proved. Properties of block diagonal matrices and their sign patterns are examined. It is shown that all 4 × 4 full sign patterns allow J-orthogonality. Important tools in this analysis are Theorem 2.2 on the exchange operator and Theorem 3.2 on the characterization of J-orthogonal matrices in the paper “J-orthogonal matrices: properties and generation", SIAM Review 45 (3) (2003), 504-519, by Higham. As a result, it follows that for n ≤4 all n×n full sign patterns allow a J-orthogonal matrix as well as a G-matrix. In addition, the 3 × 3 sign patterns of the J-orthogonal matrices which have zero entries are characterized.

The non-central Wishart distribution

1955 ◽  
Vol 229 (1178) ◽  
pp. 364-366 ◽  
Keyword(s):  
Orthogonal Group ◽  
Wishart Distribution

The distribution is obtained in terms of an integral over the orthogonal group which is evaluated in the following paper.

On a class of determinantal primals and their multiple loci

1951 ◽  
Vol 47 (2) ◽  
pp. 286-298 ◽  
Author(s):  
L. S. Goddard
Keyword(s):  
Bilinear Form ◽  
Orthogonal Group ◽  
Good Deal ◽  
Parametric Representation ◽  
Geometrical Approach ◽  
Orthogonal Matrices ◽  
Geometrical Properties ◽  
Multiple Loci ◽  
Algebraic Problem ◽  
The Subject

In this paper I investigate some geometrical properties of a system of primals which arose a few years ago in the study of a purely algebraic problem: to parametrize completely the group of automorphic transformations of a given bilinear form. This problem is classical, and there exists a large literature on the subject, but the algebraists never succeeded in finding a complete parametrization. Indeed, the trend was to move away from those transformations not covered by the known parametrization; and Weyl, for example, writing about the orthogonal group in his book on the Classical Groups remarks ‘unfortunately Cayley's parametric representation leaves out some of the orthogonal matrices, and a good deal of our efforts will be spent in rendering these exceptions ineffective’. In another paper I shall show how to solve this problem of complete parametrization, via a geometrical approach; but here I confine my attention to some preliminary geometrical results.

The arithmetical character of the Wishart distribution

1948 ◽  
Vol 44 (2) ◽  
pp. 295-297 ◽  
Author(s):  
Paul Lévy
Keyword(s):  
Generating Function ◽  
Joint Distribution ◽  
Random Variables ◽  
Wishart Distribution ◽  
Standard Deviations ◽  
Image Position

Let ξ and η be two independent and normal random variables, with zero means and with standard deviations each equal to ½ PutThe joint distribution of X, Y, Z is a particular case of the Wishart distribution (1). It may be defined by the generating function of its cumulants (c.g.f.)

scholarly journals Cospectral Graphs and Regular Orthogonal Matrices of Level 2

10.37236/2383 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Aida Abiad ◽  
Willem H Haemers
Keyword(s):  
Adjacency Matrix ◽  
Discrete Math ◽  
Orthogonal Matrix ◽  
Orthogonal Matrices ◽  
Switching Operation ◽  
Cospectral Graphs ◽  
New Methods ◽  
Almost All ◽  
Isomorphic Graphs ◽  
Level 2

For a graph $\Gamma$ with adjacency matrix $A$, we consider a switching operation that takes $\Gamma$ into a graph $\Gamma'$ with adjacency matrix $A'$, defined by $A'=Q^\top A Q$, where $Q$ is a regular orthogonal matrix of level $2$ (that is, $Q^\top Q=I$, $Q$1 $=$ 1, $2Q$ is integral, and $Q$ is not a permutation matrix). If such an operation exists, and $\Gamma$ is nonisomorphic with $\Gamma'$, then we say that $\Gamma'$ is semi-isomorphic with $\Gamma$. Semi-isomorphic graphs are $\mathbb {R}$-cospectral, which means that they are cospectral and so are their complements. Wang and Xu [On the asymptotic behavior of graphs determined by their generalized spectra, Discrete Math. 310 (2010)] expect that almost all pairs of nonisomorphic $\mathbb {R}$-cospectral graphs are semi-isomorphic.Regular orthogonal matrices of level $2$ have been classified. By use of this classification we work out the requirements for this switching operation to work in case $Q$ has one nontrivial indecomposable block of size $4$, $6$, $7$ or $8$. Size $4$ corresponds to Godsil-McKay switching. The other cases provide new methods for constructions of $\mathbb {R}$-cospectral graphs. For graphs with eight vertices all these constructions are carried out. As a result we find that, out of the 1166 graphs on eight vertices which are $\mathbb {R}$-cospectral to another graph, only 44 are not semi-isomorphic to another graph.

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