triangular%20matrices
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1: 13.27 Mathematical Applications
§13.27 Mathematical Applications
āŗConfluent 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
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āŗThis yields a lower triangular matrix of the form
…If we denote by the upper triangular matrix comprising the elements in (3.2.3), then we have the factorization, or triangular decomposition,
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āŗWe solve the system for , taking advantage of the existing triangular decomposition of to obtain an improved solution .
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āŗTridiagonal matrices are ones in which the only nonzero elements occur on the main diagonal and two adjacent diagonals.
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āŗThe
-norm of a matrix
is
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3: 1.3 Determinants, Linear Operators, and Spectral Expansions
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Determinants of Upper/Lower Triangular and Diagonal Matrices
āŗThe determinant of an upper or lower triangular, or diagonal, square matrix is the product of the diagonal elements . … āŗ§1.3(iv) Matrices as Linear Operators
… āŗReal symmetric () and Hermitian () matrices are self-adjoint operators on . … āŗFor Hermitian matrices is unitary, and for real symmetric matrices is an orthogonal transformation. …4: 1.2 Elementary Algebra
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Multiplication of Matrices
… āŗ§1.2(vi) Square Matrices
… āŗ is an upper or lower triangular matrix if all vanish for or , respectively. āŗEquation (3.2.7) displays a tridiagonal matrix in index form; (3.2.4) does the same for a lower triangular matrix. … āŗNorms of Square Matrices
…5: 3.4 Differentiation
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āŗFor additional formulas involving values of and on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546).
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6: 20 Theta Functions
Chapter 20 Theta Functions
…7: 3.11 Approximation Techniques
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āŗStarting with the first column , , and initializing the preceding column by , , we can compute the lower triangular part of the table via (3.11.25).
Similarly, the upper triangular part follows from the first row , , by initializing , .
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āŗIf , then can be factored into
matrices, the rows of which contain only a few nonzero entries and the nonzero entries are equal apart from signs.
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8: 9.19 Approximations
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Corless et al. (1992) describe a method of approximation based on subdividing into a triangular mesh, with values of , stored at the nodes. and 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 , at the node. Similarly for , .
9: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Application of the -function theory of Painlevé equations to random matrices: PIV, PII and the GUE.
Comm. Math. Phys. 219 (2), pp. 357–398.
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Application of the -function theory of Painlevé equations to random matrices: , , the LUE, JUE, and CUE.
Comm. Pure Appl. Math. 55 (6), pp. 679–727.
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Application of the -function theory of Painlevé equations to random matrices: , the JUE, CyUE, cJUE and scaled limits.
Nagoya Math. J. 174, pp. 29–114.
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10: Bibliography I
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The eigenvalue problem for infinite compact complex symmetric matrices with application to the numerical computation of complex zeros of and of Bessel functions of any real order
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Linear Algebra Appl. 194, pp. 35–70.
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The real roots of Bernoulli polynomials.
Ann. Univ. Turku. Ser. A I 37, pp. 1–20.
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