496 Index. Jacobian matrix, 170, 176 Jordan form, 354, 379, 380 Julia, 327 jump, 21, 474, 475

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1 Index A absolute stability, 188 absolute value, 83, 86 acceleration, 73, 478 accuracy, 183, 185, 190, 191 Adams method, 191, 192 addition formula, 87 add exponents, 9 adjacency matrix, 316, 318, 425 Airy s equation, 130 albedo, 49 amplitude, 75, 82, 111 amplitude response, 34, 77 antisymmetric, 244, 321, 349, 406 applied mathematics, 314, 421, 487 arrows, 155, 316 associative law, 219 attractor, 169, 180 augmented matrix, 230, 257, 271, 278 autocorrelation, 480 autonomous, 57, 71, 156, 157, 159 average, 434, 438 B back substitution, 212, 262 backslash, 220 backward difference, 6, 12, 245, 413 backward Euler, 187, 188 bad news, 326 balance equation, 48, 118, 314, 424 balance of forces, 118 bank, 12, 40, 485 bar, 403, 405, 409, 455, 457 basis, 283, 287, 289, 291, 295, 335, 444, 445 beam, 469 beat, 128 bell-shaped curve, 16, 189, 458 Bernoulli equation, 61 Bessel function, 364, 460, 478 better notation, 113, 124, 125 big picture, 298, 301, 304, 397 Black-Scholes, 457 block matrix, 230, 236, 418 block multiplication, 225, 226 boundary conditions, 417, 403, 409, 429, 457 boundary value problem, 403, 457, 470 box, 175 box function, 404, 437, 443, 469, 478, 488 Brauer, 179 C capacitance, 119 carbon, 46 carrying capacity, 53, 55, 61 Castillo-Chavez, 179 catalyst, 179 Cayley-Hamilton theorem, 345 cell phone, 44, 175 center, 160, 162, 173 centered difference, 6, 189 chain rule, 3, 4, 365, 368 change of variables, 362 chaos, 154, 180 characteristic equation, 90, 103, 108, 163 chebfun, 402 chemical engineering, 457 chess matrix, 309 Cholesky factorization, 400 circulant matrix, 204, 448, 486, 488 circular motion, 76, 348 closed-loop, 64

2 494 Index closest line, 384, 390 coefficient matrix, 198 cofactor, 328 column picture, 197, 205 column rank, 273, 320 column space, 252, 257, 276 column-times-row, 221, 225, 427 combination of columns, 198, 201 combination of eigenvectors, 326, 346, 353, 368, 371 commute, 219, 223 companion matrix, 163, 164, 166, 332, , 357, 366 competition, 53, 173 complete graph, 425, 426 complete solution, 1, 17, 18, 105, 106, 202, 210, 263, 272, 274 complex conjugate, 32, 87, 94, 376 complex eigenvalues, 165 complex exponential, 13, 430 complex Fourier series, 438 complex gain, 111 complex impedance, 120 complex matrix, 373 complex numbers, 32, 82 complex roots, 90, 162 complex solution, 36, 38, 39, 89 complex vector, 431 compound interest, 12, 184 computational mechanics, 369 computational science, 417, 445 concentration, 47, 179 condition number, 398 conductance matrix, 124, 382, 423, 424 conjugate transpose, 374 constant coefficients, 1, 98, 117, 430, 470, 487 constant diagonals, 482, 486, 487 constant source, 20 continuous, 153, 314 continuous interest, 44 convergence, 10, 195 convex, 73 convolution, 117, 136, Convolution Rule, 476, 480, 484, 485 Cooley-Tukey, 451 cooling (Newton s Law), 46 cosine series, 434 Counting Theorem, 265, 302, 312 Cramer s Rule, 328 critical damping, 96, 100, 115 critical point, 169, 170, 181 cubic spline, 139 Current Law, 123, 315, 316 cyclic convolution, D d Alembert, 464, 467 damped frequency, 99, 105, 113 damped gain, 113 damping, 96, 112, 118, 122 damping ratio, 99, 113, 114 dashpot, 118 data, 398, 429 DCT, 454 decay rate, 46, 435, 442, 456, 467 deconvolution, 485, 487 degree matrix, 316, 421, 425 delta function, 23, 28, 78, 97, 98, 404, 436, 437, 440, 458, 471 delta vector, 413, 445, 482 dependent, 286 dependent columns, 208 derivative rule, 141, 439, 476 determinant, 174, 227, 231, 323, 327, 329, 333, 344, 350, 399, 492 DFT, 430, 444, 448, 454, 485 diagonal matrix, 228, 395 diagonalizable, 354, 379 difference equation, 45, 52, 183, 187, 335 difference matrix, 239, 312, 402, 421 differential equation, 1, 40 diffusion, 355, 456, 457 diagonalization, 334, 397 dimension, 44, 52, 265, 283, , 302, 320 dimensionless, 34, 99, 113, 124 direction field, 156 Discrete Fourier Transform, (see DFT)

3 Index 495 discrete sines, 402, 430, 454 displacements, 124 distributive law, 219 divergence, 415 dot product, 200, 213, 247, 374 double angle, 84 double pole, 145, 472 double root, 90, 92, 101 doublet, 151 doubling time, 46, 47 driving function, 77, 112, 476 dropoff curve, 57, 62, 156 E echelon matrix, 261, 264, 265 edge, 311, 421 eigenfunction, 405, 419, 455, 459, 467 eigenvalue, 163, 322, 323, 379 eigenvalue matrix, 334 eigenvector, 166, 322, 323, 379 eigenvector matrix, 334, 360 Einstein, 464 elapsed time, 98 elimination, 209, 211, 331 elimination matrix, 223, 228, 301 empty set, 291 energy, 393, 394, 406, 408, 422, 441 energy balance, 48 energy identity, 438, 442 enzyme, 179 epidemic, 178, 179 equal roots, 90, 92, 101 equilibrium, 415 error, 184, 185, 190, 193 error function, 458 error vector, 383, 391 Euler, 315 Euler equations, 175, 182 Euler s Formula, 13, 82, 83, 449 Euler s method, 184, 185, 188, 381 even permutation, 245 exact equations, 65 existence, 153, 195 exponential, 2, 7, 10, 25, 131, 359, 366 exponential response, 104, 108, 117 F factorization, 379, 490 farad, 122 Fast Fourier Transform, 88, 449 feedback, 64 FFT, 430, 444, 445, 449, 451 fftw, 452 Fibonacci, 337, 342, 402 filter, 480 finite elements, 124, 370, 417, 428 finite speed, 463 first order, 163 flow graph, 452 football, 175, 177 force balance, 424 forced oscillation, 80, 105, 110 forward differences, 239 Four Fundamental Subspaces, 298, 301 Fourier coefficients, , 438 Fourier cosine series, 457 Fourier Integral Transform, 448 Fourier matrix, 85, 242, , 449 Fourier series, 419, 434, 437, 441, 455 Fourier sine series, 407, 432, 467 fourth order, 80, 93, 469 foxes, 171, 173 free column, 260 free variable, 262, 264, 267, 268, 272 free-free boundary conditions, 409 frequency, 31, 76, 79, 370, 466 frequency domain, 120, 145, 448, 480 frequency response, 36, 77, 430 frisbee, 175 full rank, , 279, 285, 382 function space, 291, 296, 431, 438, 480 fundamental matrix, 363, 368, 381 fundamental solution, 78, 81, 97, 117, 458 Fundamental Theorem, 5, 8, 42, 243, 302, 305, 397 G gain, 30, 33, 84, 104, 111 Gauss-Jordan, , 235, 281, 328 gene, 429

4 496 Index general solution, 278 generalized eigenvalues, 369 geometric series, 7 Gibbs phenomenon, 433, 434 gold, 152 Gompertz equation, 63 Google, 325 GPS, 464 gradient, 415, 419 graph, 311, 315, 316, 318, 414, 421 graph Laplacian, 316, 318, 422 Green s function, 136, 482, 483 greenhouse effect, 49 grid, 414, 417, 427 ground a node, 422, 424 growth factor, 24, 40 42, 51, 97, 135, 482 growth rate, 2, 40, 362 H HKenon map, 180 Hadamard matrix, 242, 341 half-life, 46 harmonic motion, 75, 76, 79 harvesting, 59, 60, 62 hat function, 467 heat equation, 407, 455, 456 heat kernel, 457, 458, 460 Heaviside, 21, 477 Henry, 122 Hermitian matrix, 374 Hertz, 76 higher order, 93, 102, 105, 107, 117, 352 Hilbert space, 431 homogeneous, 17, 103 Hooke s Law, 74, 371, 422 hyperplane, 206 I identity matrix, 200, 218 image, 484 imaginary eigenvalues, 329, 348 impedance, 39, 120, 121, 127 implicit, 67, 187 impulse, 23, 78 impulse response, 23, 24, 78, 97, 102, 117, 121, 136, 140, 150, 482 incidence matrix, 124, 311, 315, 318, 421 independence, 203 independent columns, 271, 274, 288, 320, 382, 388 independent eigenvectors, 359 independent rows, 271 inductance, 119 infection rate, 178 infinite series, 10, 13, 326, 366, 432, 455 inflection point, 54, 55 initial conditions, 2, 40, 73, 346, 457 initial values, 470, 483 inner product, 225, 321, 374, 406, 431 instability, 192 integrating factor, 19, 26, 41, 482 integration by parts, 247, 321, 406, 411, 429 interest rate, 12, 43, 485 intersection, 200, 256, 297 inverse matrix, 31, 227, 230, 482 inverse transform, 140, 444, 473, 477 invertible, 204, 212, 227, 288 isocline, 155, 158, 159 J Jacobian matrix, 170, 176 Jordan form, 354, 379, 380 Julia, 327 jump, 21, 474, 475 K key formula, 8, 19, 78, 112, 117, 135, 482 kinetic energy, 79 Kirchhoff s Current Law, 314, 422 Kirchhoff s Laws, 123, 270 Kirchhoff s Voltage Law, 313 KKT matrix, 426 kron.a; B/, 418 L l H Oopital s Rule, 43, 109 LAPACK, 241, 329 Laplace convolution, 481, 483 Laplace equation, 414, 415

5 Index 497 Laplace transform, 121, , Laplace s equation, 416, 440, 441 Laplacian matrix, 316, 318, 422 law of mass action, 179 least squares, left eigenvectors, 345 left nullspace, 298, 300 left-inverse, 227, 231, 241 length, 241 LiKenard, 181 linear combination, 198, 200, 252, 286 linear equation, 4, 17, 105, 130, 176, 346 linear shift-invariant, 459 linear time-invariant (LTI), 71, 346 linear transformation, 208 linearity, 220, 471 linearization, linearly independent, 275, 285, 287 lobster trap, 158 logistic equation, 47, 53, 62, 156, 189 loop, loop equation, 119, 120, 123 Lorenz equation, ix, 154, 180 Lotka-Volterra, 172 M magic matrix, 208 magnitude, 112 magnitude response, 34, 77 Markov matrix, 324, 326, 330, 379 mass action, 179 mass matrix, 369, 378 Mathematica, 193, 467 mathematical finance, 457 MATLAB, 190, 327, 369, 444, 451, 486 The single heading Matrix indexes the active life of linear algebra. Matrix 1; 2; 1, 245, 413, 454 adjacency, 316 antisymmetric, 349, 379 augmented, 230, 271, 278 circulant, 486, 488 companion, 163, 352, 357 complex, 373 difference, 239, 312, 402, 421, echelon, 264 eigenvalue, 334 eigenvector, 334, 360 elimination, 223, 228, 301 exponential, 14, 359, 367 factorizations, 379, 490 Fourier, 85, 242, 444, 447, 449 fundamental, 363 Hadamard, 242, 341 Hermitian, 374 identity, 200, 218 incidence, 124, 311, 312, 315, 421 inverse, 227, 230 invertible, 204, 212, 227, 288 Jacobian, 170, 176 KKT, 426, 428 Laplacian, 316, 318, 422 Markov, 324, 330 orthogonal, 237, 246, 373 permutation, 240, 245, 297, 449 positive definite, 369, 382, 393 projection orthogonal, 237, 241, 246, 331, 373, 378, 379, 386, 391 rank one, 303, 379, 398 rectangular, 382 reflection, 246 rotation, 328 saddle-point, 426, 428 second difference, 412 semidefinite, 395, 409, 411 similar, 362, 367, 380 singular, 201, 323, 325, 500 skew-symmetric, 379 sparse, 222 stable, 349 stiffness, 124, 369, 382 symmetric, 165, 237, 372, 406 Toeplitz, 480, 482 tridiagonal, 379, 454 unitary, 374

6 498 Index matrix multiplication, , 248 mean, 389, 392 mechanics, 74 mesh, 418 Michaelis-Menten, 179 minimum, 401 model problem, 40, 115, 371, 421 modulus, 32, 83 multiplication, 201, 218, 479 multiplicity, 93, 340 multiplier, 209, 213, 224 multistep method, 191 N natural frequency, 77, 99, 102, 466 network, , 414, 423, 424 neutral stability, 165, 336, 350 Newton s Law, 46, 73, 239, 370 Newton s method, 6, 180 nodal analysis, 123 node, 311, 421 nondiagonalizable, 336, 339, 343, 380 nonlinear equation, 1, 53, 171 nonlinear oscillation, 71 norm, 397, 398 normal distribution, 458 normal equations, 384, 386 normal modes, 370 Nth order equation, 107, 117 null solution, 17, 18, 78, 92, 103, 106, 113, 202 nullity, 265 nullspace, 259 number of solutions, 280 O ODE 45, 190, 192 off-diagonal ratios, 226 Ohm s Law, 39, 122, 422, 423, 425 one-way wave, 463, 468 open-loop, 64 operation count, 452 optimal control, 478 order of accuracy, 185, 190, 191 orthogonal basis, 396, 431, 446, 447 orthogonal eigenvectors, 238, 372 orthogonal functions, 321, 402, 432 orthogonal matrix, 237, 246, 373, 378 orthogonal subspace, 304 orthonormal basis, 395, 397, 438 orthonormal columns, 241, 395 oscillation, 74, 75 oscillation equation, 369 overdamping, 96, 100, 102 overshoot (Gibbs), 433, 434 P PF2, 62, 142, 149, 472 PF3, 143, 149, 472 parabolas, 91, 96 parallel, 122, 127 partial differential equation, (see PDE) partial fractions, 56, 62, , 474 partial sums, 436 particular solution, 17, 18, 41, 106, 202, 272, 274, 276 PDE, 414, 455, 463 peak time, 113, 128 pendulum, 71, 81, 181 period, 76, 162, 442 periodic, 172 permutation matrix, 240, 245, 297, 449 perpendicular, 200, 242, 389, 433, 434 perpendicular eigenvectors, 380 perpendicular subspaces, 310 phase angle, 32, 80 phase lag, 30, 33, 75, 81, 104, 112 phase line, 169 phase plane, 59, 348 phase response, 77 pictures, 152, 161 pivot, 209, 211, 224, 232, 399 pivot column, 260, 262, 288, 292 pivot variable, 262, 268 plane, 200, 206, 256 Pluto, 154 point source, 23, 457, 458 point-spread function, 484 Poisson s equation, 415 polar angle, 38, 83

7 Index 499 polar form, 30, 32, 84, 110, 112, 121, 243, 416, 429, 447 poles, 100, 129, 140, polynomial, 131 Pontryagin, 478 population, 47, 55, 61, 63 positive definite, 369, 382, 394, positive definite matrix, 369, 382, 393 positive semidefinite, 409, 411 potential energy, 79 powers, 220, 325, 338 practical resonance, 126 predator-prey, 171, 173, 179 prediction-correction, 190 present value, 51 principal axis, 373 Principal Component Analysis, 398, 429 probability, 458 product integral, 381 product of pivots, 327, 500 product rule, 8 projection, 384, , 391 projection matrix, 246, 331, 379, 386, 391 pulse, 389, 390 Python, 327 Q quadratic formula, 90 quiver, 154 R rabbits, 171, 173 radians, 76 radioactive decay, 45 ramp function, 23, 98, 404, 405, 477 ramp response, 129 rank, 265, 271, 275, 299 rank of AB, 309 rank one matrix, 303, 379, 398 rank theorem, 320 Rayleigh quotient, 429 reactance, 121 real eigenvalues, 165, 238, 372 real roots, 90, 161 real solution, 31, 111 rectangular form, 110, 111 rectangular matrix, 382 recursion, 452, 453 red lights, 478 reflection matrix, 246, 379 relativity, 464 relaxation time, 46 repeated eigenvalues, 335, 336, 352, 380 repeated roots, 90, 92, 101, 352 repeating ramp, 434 resistance, 119, 424 resonance, 26, 27, 29, 79, 82, 108, 109, 114, 116, 132, 137, 361 response, 77 reverse order, 228, 237, 247 right triangle, 129, 383 right-inverse, 227, 231, 232 RLC loop, 39, 118, 119, 122 roots, 101, 108, 129 roots of z N D 1, 447 rotation matrix, 328 row exchange, 211, 215, 241 row picture, 196, 197, 213 row space, 287, 321 rref.a/, 261, 263, 265, 266, 282 Runge-Kutta, 16, S S-curve, 54, 64, 156 saddle, 161, 168, 172, 176, 399, 426 saddle-point matrix, 428 SciPy, 193 second difference, 239, 245, 407, 412, 413 semidefinite, 395, 409 separable, 56, 65 separation of variables, 419, 420, 456, 459, 460, 466 shift, 439 shift invariance, 98, 459, 480, 482, 487 shift rule for transform, 475 sign reversal, 500 similar matrix, 362, 367, 380 Simpson s Rule, 194 sines and cosines, 437 singular matrix, 201, 204, 214, 323, 500

8 500 Index singular value, 395, 397, 402 Singular Value Decomposition, 395 singular vector, 382 sink, 17, 161 sinusoid, 19, 30, 34 sinusoidal identity, 35, 37, 112 SIR model, 178 six pictures, 161, 170 skew-symmetric, 378 smoothness, 435 solution curve, 153 Solution Page, 117 solvable, 253, 255, 276, 309 source, 17, 19, 40, 161 span, 254, 258, 283, 286, 294 sparse matrices, 222 special inputs, 131, 139 special solution, 259, 263, 300 spectral theorem, 373, 380 speed of light, 464 spike, 23, 404, 435, 436 spiral, 33, 86, 88, 95, 160 spiral sink, 162 spring, 74, 119 square root, 394 square wave, 433, 435, 441, 456 stability, 49, 58 60, 186, 187 stability limit, 187, 194 stability line, 58, 169 stability test, , 174, 187, 336, 350 stable, 160, 168, 349, 372 standing wave, 465 starting value (initial condition), 2, 9 state space, 127 statistics, 398, 458 steady state, 21, 49, 53, 58, 154, 325, 354 Stefan-Boltzmann Law, 49, 63 step function, 21, 23, 474, 475, 478, 489 step response, 22, 81, 97, 102, stepsize, 183 stiff equation, 186 stiff system, 192 stiffness, 118, 468 stiffness matrix, 124, 369, 382 stock prices, 457 straight line, 383 subspace, , 254, 256, 294 sudoku matrix, 208 sum of spaces, 258 sum of squares, 383, 385 superposition, 8, 346, 460 SVD, 243, 379, 382, , 429 switch, 22 symmetric and orthogonal, 243, 375 symmetric matrix, 237, 238, 290, 372, 406 symmetry, 468 system, 163, 196, 322 T Table of eigenvalues, 379 Table of Rules, 476 Table of Transforms, 146, 471 tangent, 75, 80, 155 tangent line, 6, 183 tangent parabola, 7, 190 Taylor series, 7, 10, 14, 16, 184 temperature, 46, 440, 455, 459 test grades, 392 three steps, 338, 346, 366 time constant, 100 time domain, 120, 127 time lag, 81 time-varying, 364, 368, 381 Toeplitz matrix, 480, 482 Toomre, 177 trace, 174, 328, 329, 333, 344, 350, 381 transfer function, 104, 121, 430, 477, 481 transient, 27, 103 tree, 315 triangular matrix, 213, 236, 293, 490, 492 tridiagonal matrix, 231, 245, 379, 407, 454 tumbling box, 175, 177, 182 U underdamping, 96, 100, 102, 117 undetermined coefficients, 117, uniqueness, 153, 287 unit circle, 33, 84, 85, 94, 447 unit vector, 331

9 Index 501 unitary matrix, 374 units, 44, 52, 456 unstable, 49, 53, 165 upper triangular, 209, 213 V variable coefficient, 1, 42, 130 variance, 389, 392, 398, 429 variation of parameters, 41, 43, 130, , 138, 482 vector, 163, 198, 199, 249, 250 vector space, 249, 250, 296, 319 very particular, 26, 27, 117, 144 violin, 465, 469 Voltage Law, 123, 315, 316 voltage source, 423 W wave equation, , 469 weighted Laplacian, 422, 425 weighted least squares, 388, 389 Wikipedia, 242, 429 Wolfram Alpha, 193 Wronskian, 134, 135, 363, 381 Index of Symbols A D LU, 412, 490 A D QR, 490 A D QS, 429 A D U V T, 379, 395, 398 A D VƒV 1, 334, 338 A T A, 238, 274, 310, 382, 392, 415, 421 A T CA, 389, 401, 414, 423, 425 A D A T, 411 K2D, 417, 418 K D A T CA, 408, 421, 422 P.D/, 108, 117 Q, 237 S D LDL T, 400 S D QƒQ T, 373 S?, 305 C.A/ and N.A/, 253, 259 R n and C n, 249 Z zerocline, 156 zeta, 99, 113