Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...
Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ... Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...
Appendix D 273 Appendix D: Matrix methods This appendix has been taken from the “Introduction to Molecular Spectroscopy” script, not all subsections apply to chemical kinetics. D.1 Definition I Matrices: A matrix is a rectangular, × dimensional array of numbers : A = ⎛ ⎜ ⎝ ⎞ 11 12 13 1 21 22 23 2 ⎟ . . . . ⎠ 1 2 3 (D.1) The are called the matrix elements. We will need to consider only × dimensional square matrices, which we can think of as a collection of column vectors a : ⎛ ⎞ 11 12 1 A = ⎜ 21 22 2 ⎟ ⎝ . . . ⎠ (D.2) 1 2 I Notation: Matrices will be denoted by bold symbols: A (D.3) I Matrix manipulation by computer programs: Matrix manipulations are carried out efficiently using computer programs (see, e.g., Press1992) or with symbolic algebra programs. 56 56 The most commmon symbolic math programs are MathCad, MuPad, Maple, andMathematica. MathCad and Mathematica are available in the PC lab.
Appendix D 274 D.2 Special matrices and matrix operations I Identity matrix: I = 1 I = 1 = ⎛ ⎜ ⎝ 10 0 01 0 . . . 00 1 ⎞ ⎟ ⎠ (D.4) We frequently use the Kronecker : = (D.5) where ½ 1 for = = 0 for 6= (D.6) I Diagonal matrices: ⎛ ⎜ ⎝ ⎞ 11 0 0 22 0 ⎟ . . . . ⎠ 0 0 (D.7) I Block diagonal matrices: ⎛ ⎞ 11 12 0 ⎜ 21 22 0 ⎟ ⎝ . . . . ⎠ 0 0 (D.8) I Complex conjugate (c.c.) of a matrix: A ∗ To obtain the complex conjugate A ∗ of the matrix A, change to −: ( ∗ ) =( ) ∗ (D.9) I Transpose of a matrix: To obtain the transpose A of the matrix A, interchange rows and colums: ¡ ¢ =( ) (D.10)
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- Page 277 and 278: Appendix B 262 Appendix A: Useful p
- Page 279 and 280: Appendix B 264 The Marquardt-Levenb
- Page 281 and 282: Appendix C 266 • Convolution of t
- Page 283 and 284: Appendix C 268 C.2 Application to t
- Page 285 and 286: Appendix C 270 C.3 Application to p
- Page 287: Appendix D 272 Final solutions for
- Page 291 and 292: Appendix D 276 I Multiplication by
- Page 293 and 294: Appendix D 278 D.4 Coordinate trans
- Page 295 and 296: Appendix D 280 D.5 Systems of linea
- Page 297 and 298: Appendix D 282 D.6 Eigenvalue equat
- Page 299 and 300: Appendix E 284 Secular equation: de
- Page 301 and 302: Appendix E 286 Appendix E: Laplace
- Page 303 and 304: Appendix F 288 Appendix F: The Gamm
- Page 305 and 306: Appendix G 290 Appendix G: Ergebnis
- Page 307 and 308: Appendix G 292 I Anschauliche Bedeu
- Page 309 and 310: Appendix G 294 Ergebnis: = 1 X
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- Page 313 and 314: Appendix G 298 G.4 Mikrokanonische
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Appendix D 273<br />
Appendix D: Matrix methods<br />
This appendix has been taken from the “Introduction to Molecular Spectroscopy” script,<br />
not all subsections apply to chemical kinetics.<br />
D.1 Definition<br />
I Matrices: A matrix is a rectangular, × dimensional array of numbers :<br />
A =<br />
⎛<br />
⎜<br />
⎝<br />
⎞<br />
11 12 13 1<br />
21 22 23 2<br />
⎟<br />
. . . . ⎠<br />
1 2 3 <br />
(D.1)<br />
The are called the matrix elements.<br />
We will need to consider only × dimensional square matrices, which we can think<br />
of as a collection of column vectors a :<br />
⎛<br />
⎞<br />
11 12 1<br />
<br />
A = ⎜ 21 22 2<br />
⎟<br />
⎝ . . . ⎠<br />
(D.2)<br />
1 2 <br />
I Notation: Matrices will be denoted by bold symbols:<br />
A<br />
(D.3)<br />
I Matrix manipulation by computer programs: Matrix manipulations are carried<br />
out efficiently using computer programs (see, e.g., Press1992) or with symbolic algebra<br />
programs. 56<br />
56 The most commmon symbolic math programs are MathCad, MuPad, Maple, andMathematica.<br />
MathCad and Mathematica are available in the PC lab.
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