# triangular decomposition

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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. … …
##### 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 $\mathbf{x}+\delta\mathbf{x}$. … In the case that the orthogonality condition is replaced by $\mathbf{S}$-orthogonality, that is, $\mathbf{v}_{j}^{\rm T}\mathbf{S}\mathbf{v}_{k}=\delta_{j,k}$, $j,k=1,2,\ldots,n$, for some positive definite matrix $\mathbf{S}$ with Cholesky decomposition $\mathbf{S}=\mathbf{L}^{\rm T}\mathbf{L}$, then the details change as follows. …
##### 3: 16.24 Physical Applications
The $\mathit{3j}$ symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …
##### 4: 26.13 Permutations: Cycle Notation
For the example (26.13.2), this decomposition is given by ${\left(1,3,2,5,7\right)}{\left(6,8\right)}={\left(1,3\right)}{\left(2,3\right)% }{\left(2,5\right)}{\left(5,7\right)}{\left(6,8\right)}.$Again, for the example (26.13.2) a minimal decomposition into adjacent transpositions is given by ${\left(1,3,2,5,7\right)}{\left(6,8\right)}={\left(2,3\right)}\*{\left(1,2% \right)}\*{\left(4,5\right)}{\left(3,4\right)}{\left(2,3\right)}{\left(3,4% \right)}{\left(4,5\right)}{\left(6,7\right)}{\left(5,6\right)}{\left(7,8\right% )}\*{\left(6,7\right)}$: $\mathop{\mathrm{inv}}({\left(1,3,2,5,7\right)}{\left(6,8\right)})=11$.
##### 5: 9.19 Approximations
• Corless et al. (1992) describe a method of approximation based on subdividing $\mathbb{C}$ into a triangular mesh, with values of $\operatorname{Ai}\left(z\right)$, $\operatorname{Ai}'\left(z\right)$ stored at the nodes. $\operatorname{Ai}\left(z\right)$ and $\operatorname{Ai}'\left(z\right)$ 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 $\operatorname{Ai}\left(z\right)$, $\operatorname{Ai}'\left(z\right)$ at the node. Similarly for $\operatorname{Bi}\left(z\right)$, $\operatorname{Bi}'\left(z\right)$.

• ##### 6: 19.14 Reduction of General Elliptic Integrals
The last reference gives a clear summary of the various steps involving linear fractional transformations, partial-fraction decomposition, and recurrence relations. It then improves the classical method by first applying Hermite reduction to (19.2.3) to arrive at integrands without multiple poles and uses implicit full partial-fraction decomposition and implicit root finding to minimize computing with algebraic extensions. …
##### 7: 5.19 Mathematical Applications
By decomposition into partial fractions (§1.2(iii)) …
##### 8: 1.3 Determinants, Linear Operators, and Spectral Expansions
###### Determinants of Upper/Lower Triangular and Diagonal Matrices
The determinant of an upper or lower triangular, or diagonal, square matrix $\mathbf{A}$ is the product of the diagonal elements $\det(\mathbf{A})=\prod_{i=1}^{n}a_{ii}$. …
##### 9: Bibliography O
• G. E. Ordóñez and D. J. Driebe (1996) Spectral decomposition of tent maps using symmetry considerations. J. Statist. Phys. 84 (1-2), pp. 269–276.
• ##### 10: 1.2 Elementary Algebra
$\mathbf{A}$ is an upper or lower triangular matrix if all $a_{ij}$ vanish for $i>j$ or $i, respectively. Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix. …