Index
A
absolute stability, 189
absolute value, 83, 86
acceleration, 73, 478
accuracy, 184, 185, 190, 191
Adams method, 192, 193
add exponents, 9
addition formula, 87
adjacency matrix, 318, 320, 427
Airy’s equation, 130
albedo, 49
amplitude, 75, 82, 111
amplitude response, 34, 77
antisymmetric, 245, 323, 352, 409
applied mathematics, 316, 423, 487
arrows, 156, 318
associative law, 220
attractor, 170, 181
augmented matrix, 231, 259, 273, 280
autocorrelation, 480
autonomous, 57, 71, 157, 158, 160
average, 436, 440
B
back substitution, 213, 264
backslash, 221
backward difference, 6, 12, 246, 415
backward Euler, 188, 189
bad news, 329
balance equation, 48, 118, 316, 424
balance of forces, 118
bank, 12, 40, 485
bar, 406, 408, 412, 455, 457
basis, 285, 289-291, 296, 338, 446, 447
beam, 469
beat, 128
bell-shaped curve, 16, 190, 458
Bernoulli equation, 61
Bessel function, 367, 460, 478
better notation, 113, 124, 125
big picture, 300, 303, 306, 400
Black-Scholes, 457
block matrix, 231, 237, 420
block multiplication, 226, 227
boundary conditions, 406, 411, 431, 457
boundary value problem, 406, 457, 470
box, 176
box function, 407, 439, 445, 469, 478, 488
Brauer, 180
C
capacitance, 119
carbon, 46
carrying capacity, 53, 55, 61
Castillo-Chavez, 180
catalyst, 180
Cayley-Hamilton theorem, 348
cell phone, 44, 176
center, 161, 163, 174
centered difference, 6, 190
chain rule, 3, 4, 368, 371
change of variables, 365
chaos, 155, 181
characteristic equation, 90, 103, 108, 164
chebfun, 405
chemical engineering, 457
chess matrix, 311
Cholesky factorization, 403
circulant matrix, 205, 449, 486, 488
circular motion, 76, 351
closed-loop, 64
493
494
closest line, 387, 393
coefficient matrix, 199
cofactor, 331
column picture, 198, 206
column rank, 275, 322
column space, 254, 259, 278
column-times-row, 222, 226, 429
combination of columns, 199, 202
combination of eigenvectors, 329, 349,
356, 371, 374
commute, 221, 224
companion matrix, 164, 165, 167, 335,
354–356, 360, 369
competition, 53, 174
complete graph, 427, 428
complete solution, 1, 17, 18, 105, 106,
203, 211, 265, 274, 276
complex conjugate, 32, 87, 94, 379
complex eigenvalues, 166
complex exponential, 13, 432
complex Fourier series, 440
complex gain, 111
complex impedance, 120
complex matrix, 376
complex numbers, 31-33, 82-89
complex roots, 90, 163
complex solution, 36, 38, 39, 89
complex vector, 433
compound interest, 12, 185
computational mechanics, 372
computational science, 419, 447
concentration, 47, 180
condition number, 401
conductance matrix, 124, 385, 425, 426
conjugate transpose, 377
constant coefficients, 1, 98, 117, 432,
470, 487
constant diagonals, 482, 486, 487
constant source, 20
continuous, 154, 358
continuous interest, 44
convergence, 10, 196
convex, 73
convolution, 117, 136, 479-489
Index
Convolution Rule, 476, 480, 484, 485
Cooley-Tukey, 451
cooling (Newton’s Law), 46
cosine series, 436
Counting Theorem, 267, 304, 314
Cramer’s Rule, 331
critical damping, 96, 100, 115
critical point, 170, 171, 182
cubic spline, 139
Current Law, 123, 317, 318
cyclic convolution, 485–487
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, 401, 431
decay rate, 46, 437, 444, 456, 467
deconvolution, 485, 487
degree matrix, 318, 427, 429
delta function, 23, 28, 78, 97, 98, 407,
438, 439, 442, 458, 471
delta vector, 415, 447, 482
dependent, 288
dependent columns, 209
derivative rule, 141, 441, 476
determinant, 175, 228, 232, 326, 330,
332, 336, 347, 353, 402, 492
DFT, 432, 446, 449, 454, 485
diagonal matrix, 229, 398
diagonalizable, 363, 382
difference equation, 45, 52, 184, 188, 338
difference matrix, 240, 314, 405, 423
differential equation, 1, 40, 349
diffusion, 358, 456, 457
diagonalization, 337, 400
dimension, 44, 52, 267, 285, 291–293,
304, 322
dimensionless, 34, 99, 113, 124
direction field, 157
Discrete Cosine Transform (DCT), 454
Discrete Fourier Transform, (see DFT)
495
Index
discrete sines, 405, 432, 454
displacements, 124
distributive law, 220
divergence, 417
dot product, 201, 214, 248, 377
double angle, 84
double pole, 145, 472
double root, 91, 92, 101
doublet, 151
doubling time, 46, 47
driving function, 77, 112, 476
dropoff curve, 57, 62, 157
E
echelon matrix, 263, 266, 267
edge, 313, 423
eigenfunction, 408, 421, 455, 459, 467
eigenvalue, 164, 325, 326, 382
eigenvalue matrix, 337
eigenvector, 167, 325, 326, 382
eigenvector matrix, 337, 363
Einstein, 464
elapsed time, 98
elimination, 210, 212, 334
elimination matrix, 224, 229, 303
empty set, 293
energy, 396, 397, 409, 411, 424, 443
energy balance, 48
energy identity, 440, 444
enzyme, 180
epidemic, 179, 180
equal roots, 90, 92, 100
equilibrium, 417
error, 185, 186, 191, 193
error function, 458
error vector, 386, 394
Euler, 317
Euler equations, 176, 183
Euler’s Formula, 13, 82, 83, 450
Euler’s method, 185, 186, 189, 384
even permutation, 246
exact equations, 65
existence, 154, 196
exponential, 2, 7, 10, 25, 131, 362, 369
exponential response, 104, 108, 117
F
factorization, 382, 490
farad, 122
Fast Fourier Transform, (see FFT)
feedback, 64
FFT, 88, 432, 446, 447, 450, 451
fftw, 452
Fibonacci, 340, 345, 405
filter, 480
finite elements, 124, 373, 419, 430
finite speed, 463
first order, 164
flow graph, 452
football, 176, 178
force balance, 426
forced oscillation, 80, 105, 110
forward differences, 240
Four Fundamental Subspaces, 300, 303
Fourier coefficients, 435–437, 440
Fourier cosine series, 457
Fourier Integral Transform, 449
Fourier matrix, 85, 243, 446-448, 450
Fourier series, 419, 436, 439, 443, 455
Fourier sine series, 410, 434, 467
fourth order, 80, 93, 469
foxes, 172, 174
free column, 262
free variable, 262, 266, 269, 270, 274
free-free boundary conditions, 412
frequency, 31, 76, 79, 373, 466
frequency domain, 120, 145, 449, 480
frequency response, 36, 77, 432
frisbee, 176
full rank, 275-277, 281, 287, 385
function space, 293, 298, 433, 440, 480
fundamental matrix, 366, 371, 384
fundamental solution, 78, 81, 97, 117, 458
Fundamental Theorem, 5, 8, 42, 244,
304, 307, 400
G
gain, 30, 33, 84, 104, 111
Gauss-Jordan, 230–232, 236, 283, 331
gene, 431
496
general solution, 280
generalized eigenvalues, 372
geometric series, 7
Gibbs phenomenon, 435, 436
gold, 153
Gompertz equation, 63
Google, 328
GPS, 464
gradient, 417, 421
graph, 313, 317, 318, 320, 416, 423
graph Laplacian, 316, 318, 423
Green’s function, 136, 482, 483
greenhouse effect, 49
grid, 416, 419, 429
ground a node, 424, 426
growth factor, 24, 40–42, 51, 97, 135, 482
growth rate, 2, 40, 364
H
HKenon map, 181
Hadamard matrix, 243, 344
half-life, 46
harmonic motion, 75, 76, 79
harvesting, 59, 60, 62
hat function, 467
heat equation, 410, 455, 456
heat kernel, 457, 458, 460
Heaviside, 21, 477
Henry, 122
Hermitian matrix, 377
Hertz, 76
higher order, 93, 102, 105, 107, 117, 355
Hilbert space, 433
homogeneous, 17, 103
Hooke’s Law, 74, 374, 424
hyperplane, 207
I
identity matrix, 201, 219
image, 484
imaginary eigenvalues, 331, 351
impedance, 39, 120, 121, 127
implicit, 67, 188
impulse, 23, 78
Index
impulse response, 23, 24, 78, 97, 102,
117, 121, 136, 140, 150, 482
incidence matrix, 124, 313, 317, 320, 423
independence, 204
independent columns, 273, 276, 290,
322, 385, 391
independent eigenvectors, 362
independent rows, 273
inductance, 119
infection rate, 179
infinite series, 10, 13, 329, 369, 434, 455
inflection point, 54, 55
initial conditions, 2, 40, 73, 349, 457
initial values, 470, 483
inner product, 226, 323, 377, 409, 433
instability, 193
integrating factor, 19, 26, 41, 482
integration by parts, 248, 323, 409, 413, 431
interest rate, 12, 43, 485
intersection, 201, 258, 299
inverse matrix, 31, 228, 231, 482
inverse transform, 140, 446, 473, 477
invertible, 205, 213, 228, 290
isocline, 156, 159, 160
J
Jacobian matrix, 171, 177
Jordan form, 357, 382, 383
Julia, 330
jump, 21, 474, 475
K
key formula, 8, 19, 78, 112, 117, 135, 482
kinetic energy, 79
Kirchhoff’s Current Law, 316, 424
Kirchhoff’s Laws, 123, 272
Kirchhoff’s Voltage Law, 315
KKT matrix, 428
kron .A; B/, 420
L
l’HOopital’s Rule, 43, 109
LAPACK, 242, 332
Laplace convolution, 481, 483
Laplace equation, 416, 417
497
Index
Laplace transform, 121, 141-151, 470-478
Laplace’s equation, 418, 442, 443
Laplacian matrix, 318, 320, 424
law of mass action, 180
least squares, 385–387
left eigenvectors, 348
left nullspace, 300, 302
left-inverse, 228, 232, 242
length, 242
LiKenard, 182
linear combination, 199, 201, 254, 288
linear equation, 4, 17, 105, 134, 177, 349
linear shift-invariant, 459
linear time-invariant (LTI), 71, 349
linear transformation, 209
linearity, 221, 471
linearization, 172-179
linearly independent, 277, 287, 289
lobster trap, 159
logistic equation, 47, 53, 62, 157, 190
loop, 315–317
loop equation, 119, 123, 127
Lorenz equation, ix, 154, 181
Lotka-Volterra, 173
M
magic matrix, 209
magnitude, 112
magnitude response, 34, 77
Markov matrix, 327, 329, 333, 382
mass action, 180
mass matrix, 372, 381
Mathematica, 194, 467
mathematical finance, 457
MATLAB, 191, 332, 372, 447, 451, 486
The single heading “Matrix” indexes
the active life of linear algebra.
Matrix
. 1 2 1/, 246, 415, 454
adjacency, 318
antisymmetric, 352, 376
augmented, 230, 271, 278
circulant, 486, 488
companion, 164, 355, 360
complex, 376
difference, 240, 314, 405, 422
echelon, 266
eigenvalue, 337
eigenvector, 337, 363
elimination, 224, 229, 303
exponential, 14, 362, 368
factorizations, 382, 490
Fourier, 85, 243, 446, 447, 450
fundamental, 366
Hadamard, 243, 344
Hermitian, 377
identity, 201, 219
incidence, 124, 313, 314, 317, 423
inverse, 228, 231
invertible, 204, 213, 231, 290
Jacobian, 171, 177
KKT, 428
Laplacian, 318, 320, 424
Markov, 327, 333
orthogonal, 238, 247, 376
permutation, 241, 246, 299, 450
positive definite, 372, 385, 396
projection orthogonal, 238, 242, 247,
334, 376, 378, 382, 390, 394
rank one, 305, 382, 404
rectangular, 385
reflection, 247
rotation, 331
saddle-point, 428, 430
second difference, 414
semidefinite, 398, 412, 413
similar, 365, 370, 383
singular, 202, 326, 328, 492
skew-symmetric, 382
sparse, 223
stable, 352
stiffness, 124, 372, 385
symmetric, 238, 375, 409
Toeplitz, 480, 482
tridiagonal, 382, 454
unitary, 377
498
matrix multiplication, 219-223, 249
mean, 392, 395
mechanics, 74
mesh, 420
Michaelis-Menten, 180
minimum, 404
model problem, 40, 115, 374, 423
modulus, 32, 83
multiplication, 202, 219, 479
multiplicity, 93, 343
multiplier, 210, 214, 225
multistep method, 192
N
natural frequency, 77, 99, 102, 466
network, 313-323, 416, 425, 426
neutral stability, 166, 339, 352
Newton’s Law, 46, 73, 239, 370
Newton’s method, 6, 181
nodal analysis, 123
node, 313, 423
nondiagonalizable, 339, 342, 346, 383
nonlinear equation, 1, 53, 172
nonlinear oscillation, 71
norm, 400, 401
normal distribution, 458
normal equations, 387, 389
normal modes, 373
Nth order equation, 107, 117
null solution, 17, 18, 78, 92, 103, 106,
113, 203
nullity, 267
nullspace, 261
number of solutions, 282
O
ODE 45, 191, 193
off-diagonal ratios, 227
Ohm’s Law, 39, 122, 424, 425, 427
one-way wave, 463, 468
open-loop, 64
operation count, 452
optimal control, 478
order of accuracy, 186, 190, 192
orthogonal basis, 399, 433, 447, 448
Index
orthogonal eigenvectors, 239, 375
orthogonal functions, 323, 405, 434
orthogonal matrix, 238, 242, 376, 381
orthogonal subspace, 306
orthonormal basis, 398, 400, 440
orthonormal columns, 242, 397
oscillation, 74, 75
oscillation equation, 372
overdamping, 96, 100, 102
overshoot (Gibbs), 435, 436
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, 142-149, 474
partial sums, 438
particular solution, 17, 18, 41, 106, 203,
274, 276, 278
PDE, 416, 455, 466
peak time, 113, 128
pendulum, 71, 81, 182
period, 76, 163, 444
periodic, 173
permutation matrix, 241, 246, 299, 450
perpendicular, 201, 243, 389, 433, 434
perpendicular eigenvectors, 383
perpendicular subspaces, 312
phase angle, 32, 80
phase lag, 30, 33, 75, 81, 112
phase line, 170
phase plane, 59, 351
phase response, 77
pictures, 153, 162
pivot, 210, 212, 225, 233, 402
pivot column, 262, 264, 290, 294
pivot variable, 264, 270
plane, 201, 207, 258
Pluto, 155
point source, 23, 457, 458
point-spread function, 484
Poisson’s equation, 417
polar angle, 38, 83
499
Index
polar form, 30, 32, 84, 110, 112, 121,
244, 418, 431, 448
poles, 100, 129, 140, 471–473
polynomial, 131
Pontryagin, 478
population, 47, 55, 61, 63
positive definite, 372, 385, 396, 403-411
positive definite matrix, 372, 382, 396
positive semidefinite, 412, 413
potential energy, 79
powers, 221, 328, 341
practical resonance, 126
predator-prey, 172, 174, 180
prediction-correction, 191
present value, 51
principal axis, 376
Principal Component Analysis, 401, 431
probability, 458
product integral, 384
product of pivots, 330, 492
product rule, 8
projection, 387, 389–391, 394
projection matrix, 247, 334, 382, 389, 394
pulse, 392, 393
Python, 330
Q
quadratic formula, 90
quiver, 155
R
rabbits, 172, 174
radians, 76
radioactive decay, 45
ramp function, 23, 98, 407, 408, 477
ramp response, 129
rank, 267, 273, 277, 301
rank of AB, 311
rank one matrix, 305, 382, 401
rank theorem, 322
Rayleigh quotient, 431
reactance, 121
real eigenvalues, 166, 239, 375
real roots, 90, 162
real solution, 31, 111
rectangular form, 110, 111
rectangular matrix, 385
recursion, 452, 453
red lights, 478
reflection matrix, 247, 382
relativity, 464
relaxation time, 46
repeated eigenvalues, 338, 339, 355, 383
repeated roots, 90, 92, 101, 355
repeating ramp, 436
resistance, 119, 426
resonance, 26, 27, 29, 79, 82, 108, 109, 114,
116, 132, 137, 364
response, 77
reverse order, 229, 238, 248
right triangle, 129, 386
right-inverse, 228, 232, 233
RLC loop, 39, 118, 119, 122
roots, 101, 108, 129
roots of z N D 1, 448
rotation matrix, 331
row exchange, 212, 216, 242
row picture, 198, 199, 214
row space, 289, 323
rref .A/, 263, 265, 267, 268, 284
Runge-Kutta, 16, 191–193
S
S -curve, 54, 64, 157
saddle, 162, 169, 173, 177, 402, 428
saddle-point matrix, 428, 430
SciPy, 194
second difference, 240, 246, 410, 414, 415
semidefinite, 398, 412
separable, 56, 65
separation of variables, 421, 422, 456,
459, 460, 466
shift, 441
shift invariance, 98, 459, 480, 482, 487
shift rule for transform, 475
sign reversal, 492
similar matrix, 365, 370, 383
Simpson’s Rule, 195
sines and cosines, 439
singular matrix, 202, 205, 218, 326, 492
500
singular value, 398, 400, 405
Singular Value Decomposition, (see SVD)
singular vector, 385
sink, 17, 162
sinusoid, 19, 30, 34
sinusoidal identity, 35, 37, 112
SIR model, 179
six pictures, 162, 171
skew-symmetric, 381
smoothness, 437
solution curve, 154
Solution Page, 117
solvable, 255, 257, 277, 311
source, 17, 19, 40, 162
span, 256, 260, 285, 288, 296
sparse matrices, 223
special inputs, 131, 139
special solution, 261, 265, 302
spectral theorem, 376, 383
speed of light, 464
spike, 23, 407, 437, 438
spiral, 33, 86, 88, 95, 161
spiral sink, 163
spring, 74, 119
square root, 397
square wave, 435, 437, 443, 456
stability, 49, 58–60, 187, 188
stability limit, 190, 195
stability line, 58, 170
stability test, 165-170, 175, 188, 339, 353
stable, 161, 169, 352, 472
standing wave, 465
starting value (initial condition), 2, 9
state space, 127
statistics, 401, 458
steady state, 21, 49, 53, 58, 155, 328, 357
Stefan-Boltzmann Law, 49, 63
step function, 21, 23, 474, 475, 478, 489
step response, 22, 81, 97, 102, 124–128
stepsize, 184
stiff equation, 187
stiff system, 193
stiffness, 118, 468
stiffness matrix, 124, 372, 385
Index
stock prices, 457
straight line, 386
subspace, 251–254, 256, 258, 296
Sudoku matrix, 209
sum of spaces, 260
sum of squares, 386, 388
superposition, 8, 349, 460
SVD, 244, 398, 382, 385, 399-405, 431
switch, 22
symmetric and orthogonal, 244, 378
symmetric matrix, 238, 239, 292, 375, 409
symmetry, 468
system, 164, 197, 325
T
Table of Eigenvalues, 382
Table of Rules, 476
Table of Transforms, 146, 471
tangent, 75, 80, 156
tangent line, 6, 184
tangent parabola, 7, 191
Taylor series, 7, 10, 14, 16, 185
temperature, 46, 442, 455, 459
test grades, 395
three steps, 341, 349, 369
time constant, 100
time domain, 120, 127
time lag, 81
time-varying, 367, 371, 384
Toeplitz matrix, 480, 482
Toomre, 178
trace, 175, 331, 332, 336, 347, 353, 384
transfer function, 104, 121, 432, 477, 481
transient, 27, 103
tree, 317
triangular matrix, 213, 238, 293, 490, 492
tridiagonal matrix, 232, 246, 382, 410, 454
tumbling box, 176, 178, 183
U
underdamping, 96, 100, 102, 117
undetermined coefficients, 117, 130-137
uniqueness, 154, 289
unit circle, 33, 84, 85, 94, 448
unit vector, 334
501
Index
unitary matrix, 377
units, 44, 52, 456
unstable, 49, 53, 166
upper triangular, 210, 213
V
variable coefficient, 1, 42, 130
variance, 392, 395, 401, 431
variation of parameters, 41, 43, 130,
133–135, 138, 482
vector, 164, 199, 200, 251, 252
vector space, 251, 252, 298, 321
very particular, 26, 27, 117, 144
violin, 465, 469
Voltage Law, 123, 317, 318
voltage source, 425
W
wave equation, 463–466, 469
weighted Laplacian, 424
weighted least squares, 390, 392
Wikipedia, 243, 431
Wolfram Alpha, 194
Wronskian, 134, 135, 366, 384
Z
zerocline, 157
zeta, 99, 113
Index of Symbols
A D LU , 414, 490
A D QR, 490
A D QS , 431
A D U †V T , 382, 398, 401
A D VƒV 1 , 337, 341
AT A, 239, 276, 312, 385, 395, 417, 423
AT CA, 392, 404, 416, 425, 427
A D AN T , 413
K2D, 419, 420
K D AT CA, 410, 423, 424
P .D/, 108, 117
Q, 238
S D LDLT , 403
S D QƒQT , 376
S ? , 307
C .A/ and N .A/, 255, 261
Rn and Cn , 251