# 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. …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: 3.2 Linear Algebra
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 $\mathbf{A}\delta\mathbf{x}=\mathbf{r}$ for $\delta\mathbf{x}$, taking advantage of the existing triangular decomposition of $\mathbf{A}$ to obtain an improved solution $\mathbf{x}+\delta\mathbf{x}$. … 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 …
##### 3: 3.11 Approximation Techniques
Starting with the first column ${[n/0]_{f}}$, $n=0,1,2,\dots$, and initializing the preceding column by ${[n/-1]_{f}}=\infty$, $n=1,2,\dots$, we can compute the lower triangular part of the table via (3.11.25). Similarly, the upper triangular part follows from the first row ${[0/n]_{f}}$, $n=0,1,2,\dots$, by initializing ${[-1/n]_{f}}=0$, $n=1,2,\dots$. … If $n=2^{m}$, then $\boldsymbol{{\Omega}}$ can be factored into $m$ matrices, the rows of which contain only a few nonzero entries and the nonzero entries are equal apart from signs. …
##### 4: 35.1 Special Notation
All matrices are of order $m\times m$, unless specified otherwise. …
 $a,b$ complex variables. … space of all real symmetric matrices. … space of positive-definite real symmetric matrices. … $\mathbf{X}-\mathbf{T}$ is positive definite. Similarly, $\mathbf{T}<\mathbf{X}$ is equivalent. … space of orthogonal matrices. …
##### 5: 29.20 Methods of Computation
A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that $n$ has to be chosen sufficiently large. … The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices $\mathbf{M}$ given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). …
##### 6: 35.9 Applications
In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${{}_{p}F_{q}}$, with $p\leq 2$ and $q\leq 1$. … In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).
##### 7: 9.19 Approximations
• Corless et al. (1992) describe a method of approximation based on subdividing $\mathbb{C}$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, $\mathrm{Ai}'\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and $\mathrm{Ai}'\left(z\right)$ are then computed from Taylor-series expansions centered at one of the nearest nodes. The Taylor coefficients are generated by recursion, starting from the stored values of $\mathrm{Ai}\left(z\right)$, $\mathrm{Ai}'\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, $\mathrm{Bi}'\left(z\right)$.

• ##### 8: Morris Newman
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. …
##### 9: 32.14 Combinatorics
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). …
##### 10: 35.5 Bessel Functions of Matrix Argument
35.5.2 $A_{\nu}\left(\mathbf{T}\right)=A_{\nu}\left(\boldsymbol{{0}}\right)\sum_{k=0}^% {\infty}\frac{(-1)^{k}}{k!}\sum_{|\kappa|=k}\frac{1}{{\left[\nu+\frac{1}{2}(m+% 1)\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in\boldsymbol{\mathcal{S}}$.
35.5.4 $\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{\nu}A_{\nu}\left(\mathbf{S}\mathbf{X}\right)\mathrm{d}{% \mathbf{X}}=\mathrm{etr}\left(-\mathbf{S}\mathbf{T}^{-1}\right)\left|\mathbf{T% }\right|^{-\nu-\frac{1}{2}(m+1)},$ $\mathbf{S}\in\boldsymbol{\mathcal{S}}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$; $\Re\left(\nu\right)>-1$.
35.5.5 $\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}A_{\nu_{1}}\left(\mathbf{S% }_{1}\mathbf{X}\right)\left|\mathbf{X}\right|^{\nu_{1}}\*A_{\nu_{2}}\left(% \mathbf{S}_{2}(\mathbf{T}-\mathbf{X})\right)\left|\mathbf{T}-\mathbf{X}\right|% ^{\nu_{2}}\mathrm{d}{\mathbf{X}}=\left|\mathbf{T}\right|^{\nu_{1}+\nu_{2}+% \frac{1}{2}(m+1)}A_{\nu_{1}+\nu_{2}+\frac{1}{2}(m+1)}\left((\mathbf{S}_{1}+% \mathbf{S}_{2})\mathbf{T}\right),$ $\nu_{j}\in\mathbb{C}$, $\Re\left(\nu_{j}\right)>-1$, $j=1,2$; $\mathbf{S}_{1},\mathbf{S}_{2}\in\boldsymbol{\mathcal{S}}$; $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.5.6 $B_{\nu}\left(\mathbf{T}\right)=\left|\mathbf{T}\right|^{-\nu}B_{-\nu}\left(% \mathbf{T}\right),$ $\nu\in\mathbb{C}$, $\mathbf{T}\in{\boldsymbol{\Omega}}$.
35.5.8 $\int_{\mathbf{O}(m)}\mathrm{etr}\left(\mathbf{S}\mathbf{H}\right)\mathrm{d}{% \mathbf{H}}=\frac{A_{-1/2}\left(-\frac{1}{4}\mathbf{S}\mathbf{S}^{\mathrm{T}}% \right)}{A_{-1/2}\left(\boldsymbol{{0}}\right)},$ $\mathbf{S}$ arbitrary.