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Index A beat, 128 absolute stability, 188 bell-shaped curve, 16, 189, 458 absolute value, 83, 86 Bernoulli equation, 61 acceleration, 73, 478 Bessel function, 364, 460, 478 accuracy, 183, 185, 190, 191 better notation, 113, 124, 125 Adams method, 191, 192 big picture, 298, 301, 304, 397 addition formula, 87 Black-Scholes, 457 add exponents, 9 block matrix, 230, 236, 418 adjacency matrix, 316, 318, 425 block multiplication, 225, 226 boundaryconditions, 417, 403, 409, 429, 457 Airy’s equation, 130 boundary value problem, 403, 457, 470 albedo, 49 box, 175 amplitude, 75, 82, 111 box function, 404, 437, 443, 469, 478, 488 amplitude response, 34, 77 Brauer, 179 antisymmetric, 244, 321, 349, 406 applied mathematics, 314, 421, 487 C arrows, 155, 316 capacitance, 119 associative law, 219 carbon, 46 attractor, 169, 180 carrying capacity, 53, 55, 61 augmented matrix, 230, 257, 271, 278 Castillo-Chavez, 179 autocorrelation, 480 catalyst, 179 autonomous, 57, 71, 156, 157, 159 Cayley-Hamilton theorem, 345 average, 434, 438 cell phone, 44, 175 center, 160, 162, 173 B centered difference, 6, 189 back substitution, 212, 262 chain rule, 3, 4, 365, 368 backslash, 220 change of variables, 362 backward difference, 6, 12, 245, 413 chaos, 154, 180 backward Euler, 187, 188 characteristic equation, 90, 103, 108, 163 bad news, 326 chebfun, 402 balance equation, 48, 118, 314, 424 chemical engineering, 457 balance of forces, 118 chess matrix, 309 bank, 12, 40, 485 Cholesky factorization, 400 bar, 403, 405, 409, 455, 457 circulant matrix, 204, 448, 486, 488 basis, 283, 287, 289, 291, 295, 335, 444, 445 circular motion, 76, 348 beam, 469 closed-loop, 64 493 494 Index closest line, 384, 390 Convolution Rule, 476, 480, 484, 485 coefficient matrix, 198 Cooley-Tukey, 451 cofactor, 328 cooling (Newton’s Law), 46 column picture, 197, 205 cosine series, 434 column rank, 273, 320 Counting Theorem, 265, 302, 312 column space, 252, 257, 276 Cramer’s Rule, 328 column-times-row, 221, 225, 427 critical damping, 96, 100, 115 combination of columns, 198, 201 critical point, 169, 170, 181 combination of eigenvectors, 326, 346, cubic spline, 139 353, 368, 371 Current Law, 123, 315, 316 commute, 219, 223 cyclic convolution, 485–487 companion matrix, 163, 164, 166, 332, D 351–353, 357, 366 d’Alembert, 464, 467 competition, 53, 173 damped frequency, 99, 105, 113 complete graph, 425, 426 damped gain, 113 complete solution, 1, 17, 18, 105, 106, damping, 96, 112, 118, 122 202, 210, 263, 272, 274 damping ratio, 99, 113, 114 complex conjugate, 32, 87, 94, 376 dashpot, 118 complex eigenvalues, 165 data, 398, 429 complex exponential, 13, 430 DCT, 454 complex Fourier series, 438 decay rate, 46, 435, 442, 456, 467 complex gain, 111 deconvolution, 485, 487 complex impedance, 120 degree matrix, 316, 421, 425 complex matrix, 373 delta function, 23, 28, 78, 97, 98, 404, complex numbers, 32, 82 436, 437, 440, 458, 471 complex roots, 90, 162 delta vector, 413, 445, 482 complex solution, 36, 38, 39, 89 dependent, 286 complex vector, 431 dependent columns, 208 compound interest, 12, 184 derivative rule, 141, 439, 476 computational mechanics, 369 determinant, 174, 227, 231, 323, 327, computational science, 417, 445 329, 333, 344, 350, 399, 492 concentration, 47, 179 DFT, 430, 444, 448, 454, 485 condition number, 398 diagonal matrix, 228, 395 conductance matrix, 124, 382, 423, 424 diagonalizable, 354, 379 conjugate transpose, 374 difference equation, 45, 52, 183, 187, 335 constant coefficients, 1, 98, 117, 430, difference matrix, 239, 312, 402, 421 470, 487 differential equation, 1, 40 constant diagonals, 482, 486, 487 diffusion, 355, 456, 457 constant source, 20 diagonalization, 334, 397 continuous, 153, 314 dimension, 44, 52, 265, 283, 289–291, continuous interest, 44 302, 320 convergence, 10, 195 dimensionless, 34, 99, 113, 124 convex, 73 direction field, 156 convolution, 117, 136, 479-489 Discrete Fourier Transform, (see DFT) Index 495 discrete sines, 402, 430, 454 F displacements, 124 factorization, 379, 490 distributive law, 219 farad, 122 divergence, 415 Fast Fourier Transform, 88, 449 dot product, 200, 213, 247, 374 feedback, 64 double angle, 84 FFT, 430, 444, 445, 449, 451 double pole, 145, 472 fftw, 452 double root, 90, 92, 101 Fibonacci, 337, 342, 402 doublet, 151 filter, 480 doubling time, 46, 47 finite elements, 124, 370, 417, 428 driving function, 77, 112, 476 finite speed, 463 dropoff curve, 57, 62, 156 first order, 163 E flow graph, 452 echelon matrix, 261, 264, 265 football, 175, 177 edge, 311, 421 force balance, 424 eigenfunction, 405, 419, 455, 459, 467 forced oscillation, 80, 105, 110 eigenvalue, 163, 322, 323, 379 forward differences, 239 eigenvalue matrix, 334 Four Fundamental Subspaces, 298, 301 eigenvector, 166, 322, 323, 379 Fourier coefficients, 433–435, 438 eigenvector matrix, 334, 360 Fourier cosine series, 457 Einstein, 464 Fourier Integral Transform, 448 elapsed time, 98 Fourier matrix, 85, 242, 444-447, 449 elimination, 209, 211, 331 Fourier series, 419, 434, 437, 441, 455 elimination matrix, 223, 228, 301 Fourier sine series, 407, 432, 467 empty set, 291 fourth order, 80, 93, 469 energy, 393, 394, 406, 408, 422, 441 foxes, 171, 173 energy balance, 48 free column, 260 energy identity, 438, 442 free variable, 262, 264, 267, 268, 272 enzyme, 179 free-free boundary conditions, 409 epidemic, 178, 179 frequency, 31, 76, 79, 370, 466 equal roots, 90, 92, 101 frequency domain, 120, 145, 448, 480 equilibrium, 415 frequency response, 36, 77, 430 error, 184, 185, 190, 193 frisbee, 175 error function, 458 full rank, 273-275, 279, 285, 382 error vector, 383, 391 function space, 291, 296, 431, 438, 480 Euler, 315 fundamental matrix, 363, 368, 381 Euler equations, 175, 182 fundamental solution, 78, 81, 97, 117, 458 Euler’s Formula, 13, 82, 83, 449 Fundamental Theorem, 5, 8, 42, 243, Euler’s method, 184, 185, 188, 381 302, 305, 397 even permutation, 245 exact equations, 65 G existence, 153, 195 gain, 30, 33, 84, 104, 111 exponential, 2, 7, 10, 25, 131, 359, 366 Gauss-Jordan, 229–231, 235, 281, 328 exponential response, 104, 108, 117 gene, 429 496 Index general solution, 278 impulse response, 23, 24, 78, 97, 102, generalized eigenvalues, 369 117, 121, 136, 140, 150, 482 geometric series, 7 incidence matrix, 124, 311, 315, 318, 421 Gibbs phenomenon, 433, 434 independence, 203 gold, 152 independent columns, 271, 274, 288, Gompertz equation, 63 320, 382, 388 Google, 325 independent eigenvectors, 359 GPS, 464 independent rows, 271 gradient, 415, 419 inductance, 119 graph, 311, 315, 316, 318, 414, 421 infection rate, 178 graph Laplacian, 316, 318, 422 infinite series, 10, 13, 326, 366, 432, 455 Green’s function, 136, 482, 483 inflection point, 54, 55 greenhouse effect, 49 initial conditions, 2, 40, 73, 346, 457 grid, 414, 417, 427 initial values, 470, 483 ground a node, 422, 424 inner product, 225, 321, 374, 406, 431 growth factor, 24, 40–42, 51, 97, 135, 482 instability, 192 growth rate, 2, 40, 362 integrating factor, 19, 26, 41, 482 integration by parts, 247, 321, 406, 411, 429 H interest rate, 12, 43, 485 Henon map, 180 intersection, 200, 256, 297 HadamardK matrix, 242, 341 inverse matrix, 31, 227, 230, 482 half-life, 46 inverse transform, 140, 444, 473, 477 harmonic motion, 75, 76, 79 invertible, 204, 212, 227, 288 harvesting, 59, 60, 62 isocline, 155, 158, 159 hat function, 467 J heat equation, 407, 455, 456 Jacobian matrix, 170, 176 heat kernel, 457, 458, 460 Jordan form, 354, 379, 380 Heaviside, 21, 477 Julia, 327 Henry, 122 jump, 21, 474, 475 Hermitian matrix, 374 Hertz, 76 K higher order, 93, 102, 105, 107, 117, 352 key formula, 8, 19, 78, 112, 117, 135, 482 Hilbert space, 431 kinetic energy, 79 homogeneous, 17, 103 Kirchhoff’s Current Law, 314, 422 Hooke’s Law, 74, 371, 422 Kirchhoff’s Laws, 123, 270 hyperplane, 206 Kirchhoff’s Voltage Law, 313 KKT matrix, 426 I kron .A;B/, 418 identity matrix, 200, 218 image, 484 L imaginary eigenvalues, 329, 348 l’Hopital’s Rule, 43, 109 O impedance, 39, 120, 121, 127 LAPACK, 241, 329 implicit, 67, 187 Laplace convolution, 481, 483 impulse, 23, 78 Laplace equation, 414, 415 Index 497 Laplace transform, 121, 140-150, 470-478 companion, 163, 352, 357 Laplace’s equation, 416, 440, 441 complex, 373 Laplacian matrix, 316, 318, 422 difference, 239, 312, 402, 421, law of mass action, 179 echelon, 264 least squares, 382–384 eigenvalue, 334 left eigenvectors, 345 eigenvector, 334, 360 left nullspace, 298, 300 elimination, 223, 228, 301 left-inverse, 227, 231, 241 exponential, 14, 359, 367 length, 241 factorizations, 379, 490 Lienard, 181 K Fourier, 85, 242, 444, 447, 449 linear combination, 198, 200, 252, 286 fundamental, 363 linear equation, 4, 17, 105, 130, 176, 346 Hadamard, 242, 341 linear shift-invariant, 459 Hermitian, 374 linear time-invariant (LTI), 71, 346 identity, 200, 218 linear transformation, 208 incidence, 124, 311, 312, 315, 421 linearity, 220, 471 inverse, 227, 230 linearization, 171-178 invertible, 204, 212, 227, 288 linearly independent, 275, 285, 287 Jacobian, 170, 176 lobster trap, 158 KKT, 426, 428 logistic equation, 47, 53, 62, 156, 189 Laplacian, 316, 318, 422 loop, 313–315 Markov, 324, 330 loop equation, 119, 120, 123 Lorenz equation, ix, 154, 180 orthogonal, 237, 246, 373 Lotka-Volterra, 172 permutation, 240, 245, 297, 449 positive definite, 369, 382, 393 M projection orthogonal, 237, 241, 246, magic matrix, 208 331, 373, 378, 379, 386, 391 magnitude, 112 rank one, 303, 379, 398 magnitude response, 34, 77 rectangular, 382 Markov matrix, 324, 326, 330, 379 reflection, 246 mass action, 179 rotation, 328 mass matrix, 369, 378 saddle-point, 426, 428 Mathematica, 193, 467 second difference, 412 mathematical finance, 457 semidefinite, 395, 409, 411 MATLAB, 190, 327, 369, 444, 451, 486 similar, 362, 367, 380 singular, 201, 323, 325, 500 The single heading “Matrix” indexes skew-symmetric, 379 the active life of linear algebra.
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  • Explicit Inverse of a Tridiagonal (P, R)–Toeplitz Matrix

    Explicit Inverse of a Tridiagonal (P, R)–Toeplitz Matrix

    Explicit inverse of a tridiagonal (p; r){Toeplitz matrix A.M. Encinas, M.J. Jim´enez Departament de Matemtiques Universitat Politcnica de Catalunya Abstract Tridiagonal matrices appears in many contexts in pure and applied mathematics, so the study of the inverse of these matrices becomes of specific interest. In recent years the invertibility of nonsingular tridiagonal matrices has been quite investigated in different fields, not only from the theoretical point of view (either in the framework of linear algebra or in the ambit of numerical analysis), but also due to applications, for instance in the study of sound propagation problems or certain quantum oscillators. However, explicit inverses are known only in a few cases, in particular when the tridiagonal matrix has constant diagonals or the coefficients of these diagonals are subjected to some restrictions like the tridiagonal p{Toeplitz matrices [7], such that their three diagonals are formed by p{periodic sequences. The recent formulae for the inversion of tridiagonal p{Toeplitz matrices are based, more o less directly, on the solution of second order linear difference equations, although most of them use a cumbersome formulation, that in fact don not take into account the periodicity of the coefficients. This contribution presents the explicit inverse of a tridiagonal matrix (p; r){Toeplitz, which diagonal coefficients are in a more general class of sequences than periodic ones, that we have called quasi{periodic sequences. A tridiagonal matrix A = (aij) of order n + 2 is called (p; r){Toeplitz if there exists m 2 N0 such that n + 2 = mp and ai+p;j+p = raij; i; j = 0;:::; (m − 1)p: Equivalently, A is a (p; r){Toeplitz matrix iff k ai+kp;j+kp = r aij; i; j = 0; : : : ; p; k = 0; : : : ; m − 1: We have developed a technique that reduces any linear second order difference equation with periodic or quasi-periodic coefficients to a difference equation of the same kind but with constant coefficients [3].