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 $𝐔$ 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 $𝐀𝛿𝐱=𝐫$ for $𝛿𝐱$, taking advantage of the existing triangular decomposition of $𝐀$ to obtain an improved solution $𝐱+𝛿𝐱$. … ►In the case that the orthogonality condition is replaced by $𝐒$-orthogonality, that is, ${𝐯}_j^{\mathrm{T}}𝐒{𝐯}_k={\delta }_{j,k}$, $j,k=1,2,\mathrm{\dots },n$, for some positive definite matrix $𝐒$ with Cholesky decomposition $𝐒={𝐋}^{\mathrm{T}}𝐋$, then the details change as follows. …

3: 16.24 Physical Applications

… ►The $\mathit{3}j$ 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 $\left(1,3,2,5,7\right)\left(6,8\right)=\left(2,3\right)\left(1,2\right)\left(4,5\right)\left(3,4\right)\left(2,3\right)\left(3,4\right)\left(4,5\right)\left(6,7\right)\left(5,6\right)\left(7,8\right)\left(6,7\right)$: $inv(\left(1,3,2,5,7\right)\left(6,8\right))=11$.

5: 5.19 Mathematical Applications

… ►By decomposition into partial fractions (§1.2(iii)) …

6: 9.19 Approximations

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Corless et al. (1992) describe a method of approximation based on subdividing $ℂ$ into a triangular mesh, with values of $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ stored at the nodes. $\mathrm{Ai}\left(z\right)$ and ${\mathrm{Ai}}^{\prime }\left(z\right)$ 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 $\mathrm{Ai}\left(z\right)$, ${\mathrm{Ai}}^{\prime }\left(z\right)$ at the node. Similarly for $\mathrm{Bi}\left(z\right)$, ${\mathrm{Bi}}^{\prime }\left(z\right)$.

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7: 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. …

8: 1.3 Determinants, Linear Operators, and Spectral Expansions

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Determinants of Upper/Lower Triangular and Diagonal Matrices

►The determinant of an upper or lower triangular, or diagonal, square matrix $𝐀$ is the product of the diagonal elements $det(𝐀)={\prod }_{i=1}^n{a}_{ii}$. …

9: Bibliography O

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Spectral decomposition of tent maps using symmetry considerations. J. Statist. Phys. 84 (1-2), pp. 269–276.

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External Links:

ISSN 0022-4715, Document, MathReview (Makoto Mori), ZentralBlatt

Referenced by:

§24.18.

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10: 1.2 Elementary Algebra

… ► $𝐀$ is an upper or lower triangular matrix if all $a_{ij}$ vanish for $i>j$ or , respectively. ►Equation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix. …