# groups of triangular matrices

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##### 1: 13.27 Mathematical Applications

###### §13.27 Mathematical Applications

►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. The elements of this group are of the form …Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function. … …##### 2: Morris Newman

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►Newman wrote the book Matrix Representations of Groups, published by the National Bureau of Standards in 1968, and the book Integral Matrices, published by Academic Press in 1972, which became a classic.
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##### 3: 17.17 Physical Applications

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►See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys.
►Quantum groups also apply $q$-series extensively.
Quantum groups are really not groups at all but certain Hopf algebras.
They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics.
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##### 4: 15.17 Mathematical Applications

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###### §15.17(iii) Group Representations

… ►Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform (§1.14(ii)) or as a specialization of a group Fourier transform. … ►###### §15.17(v) Monodromy Groups

… ►By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …##### 5: 32.14 Combinatorics

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►Let ${S}_{N}$ be the group of permutations $\bm{\pi}$ of the numbers $1,2,\mathrm{\dots},N$ (§26.2).
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►The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994).
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##### 6: Tom H. Koornwinder

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►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC.
Books for which he has been editor or coeditor include Special Functions: Group Theoretical Aspects and Applications (with R.
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►Koornwinder has been active as an officer in the SIAM Activity Group on Special Functions and Orthogonal Polynomials.
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##### 7: 21.10 Methods of Computation

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##### 8: 3.2 Linear Algebra

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►This yields a

*lower triangular matrix*of the form …If we denote by $\mathbf{U}$ the upper triangular matrix comprising the elements ${u}_{jk}$ in (3.2.3), then we have the factorization, or*triangular decomposition*, … ►We solve the system $\mathbf{A}\mathit{\delta}\mathbf{x}=\mathbf{r}$ for $\mathit{\delta}\mathbf{x}$, taking advantage of the existing triangular decomposition of $\mathbf{A}$ to obtain an improved solution $\mathbf{x}+\mathit{\delta}\mathbf{x}$. … ►*Tridiagonal matrices*are ones in which the only nonzero elements occur on the main diagonal and two adjacent diagonals. … ►The*$p$-norm of a matrix*$\mathbf{A}=[{a}_{jk}]$ is …##### 9: 21.5 Modular Transformations

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►Let $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$, and $\mathbf{D}$ be $g\times g$
matrices with integer elements such that
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►The modular transformations form a group under the composition of such transformations, the

*modular group*, which is generated by simpler transformations, for which $\xi (\mathbf{\Gamma})$ is determinate: …##### 10: Philip J. Davis

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►Davis joined the Section as part of a distinguished group of researchers studying mathematical methods for exploiting the new computational power.
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►This immediately led to discussions among some of the project members about what might be possible, and the discovery that some interactive graphics work had already been done for the NIST Matrix Market, a publicly available repository of test matrices for comparing the effectiveness of numerical linear algebra algorithms.
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