{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 0 1 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle292" -1 256 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle291" -1 257 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle290" -1 258 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle289" -1 259 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle288" -1 260 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle287" -1 261 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle286" -1 262 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle285" -1 263 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle284" -1 264 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle283" -1 265 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle282" -1 266 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle281" -1 267 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle280" -1 268 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle279" -1 269 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle278" -1 270 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle277" -1 271 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle276" -1 272 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle275" -1 273 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle274" -1 274 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle273" -1 275 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle272" -1 276 "" 1 14 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle271" -1 277 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle270" -1 278 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle269" -1 279 "" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle268" -1 280 "" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle267" -1 281 "" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle266" -1 282 "" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle265" -1 283 "" 1 12 0 0 0 0 0 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle264" -1 284 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle263" -1 285 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle262" -1 286 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle261" -1 287 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle380" -1 288 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle260" -1 289 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle379" -1 290 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle259" -1 291 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle378" -1 292 "" 0 1 0 0 0 0 0 0 1 2 2 2 0 0 0 1 } {CSTYLE "_cstyle258" -1 293 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle377" -1 294 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle257" -1 295 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle376" -1 296 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle256" -1 297 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle375" -1 298 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle374" -1 299 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle373" -1 300 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle372" -1 301 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle371" -1 302 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle370" -1 303 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle369" -1 304 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle368" -1 305 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle367" -1 306 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle366" -1 307 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle365" -1 308 "" 0 1 0 0 0 0 0 1 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle364" -1 309 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle363" -1 310 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle362" -1 311 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle361" -1 312 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle360" -1 313 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle359" -1 314 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle358" -1 315 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle357" -1 316 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle356" -1 317 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle355" -1 318 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle354" -1 319 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle353" -1 320 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle352" -1 321 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle351" -1 322 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle350" -1 323 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle349" -1 324 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle348" -1 325 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "ParagraphStyle1" -1 326 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle347" -1 327 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle346" -1 328 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle345" -1 329 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle344" -1 330 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle343" -1 331 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle342" -1 332 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle341" -1 333 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle340" -1 334 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle339" -1 335 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle338" -1 336 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle337" -1 337 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle336" -1 338 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle335" -1 339 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle334" -1 340 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle333" -1 341 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle332" -1 342 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle331" -1 343 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle330" -1 344 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle329" -1 345 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle328" -1 346 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle327" -1 347 "" 0 1 0 0 0 0 1 0 0 2 2 2 0 0 0 1 } {CSTYLE "_cstyle20" -1 348 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle21" -1 349 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "_cstyle22" -1 350 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "_cstyle23" -1 351 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle24" -1 352 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle25" -1 353 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle26" -1 354 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle27" -1 355 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle28" -1 356 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle29" -1 357 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle30" -1 358 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle31" -1 359 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle32" -1 360 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle33" -1 361 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle34" -1 362 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle35" -1 363 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle36" -1 364 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle37" -1 365 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle38" -1 366 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle39" -1 367 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "_cstyle40" -1 368 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle18" -1 220 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle19 " -1 221 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 } 1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle20" -1 222 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle21" -1 223 1 {CSTYLE "" -1 -1 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE " _pstyle22" -1 224 1 {CSTYLE "" -1 -1 "Courier" 0 1 255 0 0 1 0 1 0 2 1 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle23" -1 225 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle24" -1 226 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle25" -1 227 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle26" -1 228 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle27" -1 229 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle28" -1 230 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pst yle29" -1 231 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle30" -1 232 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle31" -1 233 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle32" -1 234 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle33" -1 235 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle34" -1 236 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle35" -1 237 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pst yle36" -1 238 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle37" -1 239 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle38" -1 240 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 } {PSTYLE "_pstyle39" -1 241 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 220 "" 0 "" {TEXT 348 42 "Section 3.3 Eigenvalues and Ei genfunctions" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 222 " " 0 "" {TEXT 349 29 "Maple Packages in Section 3.3" }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 8 "restart;" }}}{EXCHG {PARA 223 "> " 0 " " {MPLTEXT 1 350 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 224 "" 0 " " {TEXT -1 0 "" }}}}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 " " {TEXT 351 402 "We have found that having an orthogonal family in a v ector space is useful for the purposes of representing other elements \+ of the space. If one were looking for a basis for the space, an orthog onal family would be the ideal choice. We have one method for generati ng an orthogonal family: perform the Gramm-Schmidt process. In this wo rksheet we introduce another method for getting an orthogonal family. " }}{PARA 225 "" 0 "" {TEXT 351 1 " " }}{PARA 225 "" 0 "" {TEXT 351 65 "We compute eigenvectors for self-adjoint, linear transformations. " }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 352 11 " Definition:" }{TEXT 353 35 " A function A on a vector space is " } {TEXT 354 6 "linear" }{TEXT 353 3 " if" }}{PARA 227 "" 0 "" {TEXT 355 12 " A(" }{XPPEDIT 2 0 "lambda" "6#%'lambdaG" }{TEXT 355 10 " x + y) = " }{XPPEDIT 2 0 "lambda" "6#%'lambdaG" }{TEXT 355 13 " A(x) \+ + A(y);" }}{PARA 226 "" 0 "" {TEXT 353 16 "for all numbers " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 353 17 " and all vectors \+ " }{TEXT 354 1 "x" }{TEXT 353 5 " and " }{TEXT 354 1 "y" }{TEXT 353 1 "." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 228 "" 0 "" {TEXT 356 9 "Examples." }}{PARA 226 "" 0 "" {TEXT 352 1 "1" }{TEXT 353 72 ". Let A be a square matrix defined on a finite dimensional vector space." }} {PARA 226 "" 0 "" {TEXT 352 2 "2." }{TEXT 353 27 " Let the space be C \+ ''([0, " }{XPPEDIT 18 0 "infinity;" "6#%)infinityG" }{TEXT 353 10 " )) and A(" }{TEXT 354 1 "f" }{TEXT 353 4 ") = " }{TEXT 354 1 "f" }{TEXT 353 12 " '' for any " }{TEXT 354 1 "f" }{TEXT 353 14 " in the space." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 352 11 "De finition:" }{TEXT 353 79 " A linear function A is self-adjoint on the \+ space \{ E, < *, * > \} if , for all " }{TEXT 354 1 "x" }{TEXT 353 5 " and " }{TEXT 354 1 "y" }{TEXT 353 30 " in the domain of the operator " }}{PARA 229 "" 0 "" {TEXT 357 34 " < Ax , y> = < x , Ay >. " }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 228 "" 0 "" {TEXT 356 10 " Example 3." }}{PARA 225 "" 0 "" {TEXT 351 206 "One class of self-adjoo int linear functions consists of matrices which have real number entri es and which are symmetric about the main diagonal. To be specific, we illustrate with a two dimensional example." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 "" {TEXT 351 73 "First, we choose all t he entries to be real numbers, not complex numbers." }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 19 "assume(alpha,real):" }{MPLTEXT 1 350 19 "\nassume(beta,real):" }{MPLTEXT 1 350 20 "\nassume(delta,real):" } {MPLTEXT 1 350 1 "\n" }}}{PARA 225 "" 0 "" {TEXT 351 34 "Now, we defin e a symmetric matrix." }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 39 "A:=Matrix([[alpha,beta],[beta,delta]]);" }}}{PARA 225 "" 0 "" {TEXT 351 26 "Take two vectors, X and U." }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 17 "X:=Vector([x,y]);" }{MPLTEXT 1 350 18 "\nU:=Vector( [u,v]);" }}}{PARA 225 "" 0 "" {TEXT 351 70 "Compute the dot product of AX with U and the dot product of X with AU." }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 33 "AXU:=DotProduct(Multiply(A,X),U);" }{MPLTEXT 1 350 34 "\nXAU:=DotProduct(X,Multiply(A,U));" }}}{PARA 225 "" 0 "" {TEXT 351 12 "Expand this." }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 24 "expand(AXU)=expand(XAU);" }}}{PARA 225 "" 0 "" {TEXT 351 57 "F inally, ask if the left side is equal to the right side." }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 6 "is(%);" }}}{EXCHG {PARA 224 "" 0 "" {TEXT -1 0 "" }}}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 230 "" 0 "" {TEXT 358 9 "Example 4" }}{PARA 226 "" 0 "" {TEXT 353 166 "As a s econd example, we take A to be a differential operator on a subset of \+ C ''([0, 1]) with the usual dot product. The subset of C ''([0, 1]) co nsists of functions " }{TEXT 354 1 "f" }{TEXT 353 11 " such that " } {TEXT 354 1 "f" }{TEXT 353 6 "(0) = " }{TEXT 354 1 "f" }{TEXT 353 34 " (1) = 0. The definition of A is A(" }{TEXT 354 1 "f" }{TEXT 353 4 ") = " }{TEXT 354 1 "f" }{TEXT 353 134 " ''. We establish that this transf ormation is self adjoint. To see how this is done the reader needs to \+ remember integration-by-parts:" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }} {PARA 226 "" 0 "" {TEXT 353 10 " " }{XPPEDIT 18 0 "int(diff(u ,x)*v,x = a .. b) = u(b)*v(b)-u(a)*v(a)-int(u*diff(v,x),x = a .. b);" "6#/-%$intG6$*&-%%diffG6$%\"uG%\"xG\"\"\"%\"vGF-/F,;%\"aG%\"bG,(*&-F+6 #F2F--F.6#F2F-F-*&-F+6#F1F--F.6#F1F-!\"\"-F%6$*&F+F--F)6$F.F,F-/F,;F1F 2F>" }{TEXT 353 2 " ." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 225 " " 0 "" {TEXT 351 16 "We must consider" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 353 12 " " }{XPPEDIT 18 0 "int (diff(f(x),`$`(x,2))*g(x),x = 0 .. 1)-int(f(x)*diff(g(x),`$`(x,2)),x = 0 .. 1);" "6#,&-%$intG6$*&-%%diffG6$-%\"fG6#%\"xG-%\"$G6$F.\"\"#\"\" \"-%\"gG6#F.F3/F.;\"\"!F3F3-F%6$*&-F,6#F.F3-F)6$-F56#F.-F06$F.F2F3/F.; F9F3!\"\"" }{TEXT 353 1 "." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 353 36 "Using the integration by parts with " } {TEXT 354 1 "u" }{TEXT 353 3 " = " }{TEXT 354 1 "f" }{TEXT 353 8 " '' \+ and " }{TEXT 354 1 "v" }{TEXT 353 3 " = " }{TEXT 354 1 "g" }{TEXT 353 23 ", the first integral is" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }} {PARA 226 "" 0 "" {TEXT 353 10 " " }{TEXT 354 1 "f" }{TEXT 353 6 " '(1) " }{TEXT 354 1 "g" }{TEXT 353 6 "(1) - " }{TEXT 354 1 "f " }{TEXT 353 6 " '(0) " }{TEXT 354 1 "g" }{TEXT 353 7 "(0) - " } {XPPEDIT 18 0 "int(diff(f,x)*diff(g,x),x = a .. b);" "6#-%$intG6$*&-%% diffG6$%\"fG%\"xG\"\"\"-F(6$%\"gGF+F,/F+;%\"aG%\"bG" }{TEXT 353 2 " . " }}{PARA 226 "" 0 "" {TEXT 353 51 "On the other hand, using integrati on by parts with " }{TEXT 354 1 "u" }{TEXT 353 3 " = " }{TEXT 354 1 "g " }{TEXT 353 8 " '' and " }{TEXT 354 1 "v" }{TEXT 353 3 " = " }{TEXT 354 1 "f" }{TEXT 353 36 " we have that the second integral is" }} {PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 353 10 " \+ " }{TEXT 354 1 "g" }{TEXT 353 6 " '(1) " }{TEXT 354 1 "f" } {TEXT 353 6 "(1) - " }{TEXT 354 1 "g" }{TEXT 353 6 " '(0) " }{TEXT 354 1 "f" }{TEXT 353 6 "(0) - " }{XPPEDIT 18 0 "int(diff(f,x)*diff(g,x ),x = 0 .. 1);" "6#-%$intG6$*&-%%diffG6$%\"fG%\"xG\"\"\"-F(6$%\"gGF+F, /F+;\"\"!F," }{TEXT 353 2 " ." }}{PARA 225 "" 0 "" {TEXT 351 60 "When \+ we subtract the second integral from the first, we have" }}{PARA 221 " " 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 353 10 " " } {TEXT 354 1 "f" }{TEXT 353 6 " '(1) " }{TEXT 354 1 "g" }{TEXT 353 6 "( 1) - " }{TEXT 354 1 "f" }{TEXT 353 6 " '(0) " }{TEXT 354 1 "g" }{TEXT 353 6 "(0) - " }{TEXT 354 1 "g" }{TEXT 353 6 " '(1) " }{TEXT 354 1 "f " }{TEXT 353 6 "(1) + " }{TEXT 354 1 "g" }{TEXT 353 11 " '(0) f(0)." } }{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 353 31 "Thi s last line will be zero if " }{TEXT 354 1 "f" }{TEXT 353 5 " and " } {TEXT 354 1 "g" }{TEXT 353 71 " satisfy the boundary conditions to be \+ in the domain of A, that is, if " }}{PARA 226 "" 0 "" {TEXT 353 4 " \+ " }{TEXT 354 1 "f" }{TEXT 353 10 "(1) = 0 = " }{TEXT 354 1 "f" } {TEXT 353 8 "(0) and " }{TEXT 354 1 "g" }{TEXT 353 10 "(1) = 0 = " } {TEXT 354 1 "g" }{TEXT 353 4 "(0)." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 "" {TEXT 351 266 "We have established that this A -- \+ take two derivatives, with boundary conditions -- is self adjoint. (Th e reader might observe in passing that there are other boundary condit ions that would achieve this same condition. We will see some of these in future problems.)" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 " " 0 "" {TEXT 352 11 "Definition:" }{TEXT 353 12 " The number " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 353 33 " is an eigenvalue and the vector " }{TEXT 354 1 "v" }{TEXT 353 52 " is an eigenvector o f the linear transformation A if" }}{PARA 226 "" 0 "" {TEXT 353 12 " \+ A(" }{TEXT 354 1 "v" }{TEXT 353 4 ") = " }{XPPEDIT 18 0 "lambd a;" "6#%'lambdaG" }{TEXT 353 1 " " }{TEXT 354 1 "v" }{TEXT 353 1 "." } }{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 231 "" 0 "" {TEXT 359 9 "Exam ple 5" }}{PARA 225 "" 0 "" {TEXT 351 71 "The numbers -1 and -2 are eig envalues corresponding to the eigenvectors" }}{PARA 226 "" 0 "" {TEXT 353 10 " " }{XPPEDIT 18 0 "matrix([[1], [1]]);" "6#-%'matrixG 6#7$7#\"\"\"7#F(" }{TEXT 353 6 " and " }{XPPEDIT 18 0 "matrix([[1], [ -1]]);" "6#-%'matrixG6#7$7#\"\"\"7#,$F(!\"\"" }{TEXT 353 1 " " }} {PARA 225 "" 0 "" {TEXT 351 25 "for the matrix A given as" }}{PARA 226 "" 0 "" {TEXT 353 35 " A = " } {XPPEDIT 18 0 "matrix([[-3/2, 1/2], [1/2, -3/2]]);" "6#-%'matrixG6#7$7 $,$*&\"\"$\"\"\"\"\"#!\"\"F-*&F+F+F,F-7$*&F+F+F,F-,$*&F*F+F,F-F-" }} {PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 "" {TEXT 351 55 "We u se Maple to give those eigenvalues and eigenvectors" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 35 "A:=Ma trix([[-3/2,1/2],[1/2,-3/2]]);" }}}{PARA 225 "" 0 "" {TEXT 351 277 "Ma ple produces a list. The first entry is interpreted as follows: -1 is \+ an eigenvalue of multiplicity 1 and corresponding to the eigenvector [ 1, 1]. The second entry is interpreted asfollows: -2 is an eigenvalue \+ of multiplicity 1 and corresponding to the eigenvector [-1, 1]. " }} {EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 31 "Eigenvectors(A, output=' list');" }}}{EXCHG {PARA 224 "" 0 "" {TEXT -1 0 "" }}}{PARA 221 "" 0 " " {TEXT -1 0 "" }}{PARA 232 "" 0 "" {TEXT 360 9 "Example 6" }}{PARA 226 "" 0 "" {TEXT 353 20 "Each of the numbers " }{XPPEDIT 18 0 "-pi^2; " "6#,$*$%#piG\"\"#!\"\"" }{TEXT 353 2 ", " }{XPPEDIT 18 0 "-4*Pi^2;" "6#,$*&\"\"%\"\"\"*$%#PiG\"\"#F&!\"\"" }{TEXT 353 3 " , " }{XPPEDIT 18 0 "-9*Pi^2;" "6#,$*&\"\"*\"\"\"*$%#PiG\"\"#F&!\"\"" }{TEXT 353 7 ", ..., " }{XPPEDIT 18 0 "-n^2*Pi^2;" "6#,$*&%\"nG\"\"#%#PiGF&!\"\"" } {TEXT 353 58 " ... is an eigenvalue corresponding to the eigenfunctio n " }}{PARA 226 "" 0 "" {TEXT 353 10 " " }{XPPEDIT 18 0 "sin( n*Pi*x);" "6#-%$sinG6#*(%\"nG\"\"\"%#PiGF(%\"xGF(" }}{PARA 225 "" 0 " " {TEXT 351 64 "for the linear operator of Example 4 above. We verify \+ this here." }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 18 "assume(n,i nteger):" }{MPLTEXT 1 350 19 "\nu:=x->sin(n*Pi*x);" }{MPLTEXT 1 350 19 "\nlambda:=-n^2*Pi^2;" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 31 "is(diff(u(x),x,x)=lambda*u(x));" }{MPLTEXT 1 350 12 "\nis(u(0) =0);" }{MPLTEXT 1 350 12 "\nis(u(1)=0);" }}}{EXCHG {PARA 224 "" 0 "" {TEXT -1 0 "" }}}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 361 6 "Remark" }{TEXT 353 93 ": We have presented the eigenvalue s and eigenvectors for these two linear transformations as " }{TEXT 354 14 "faits accompli" }{TEXT 353 271 ". The process for getting eige nvalues and eigenvectors for a matrix is the subject of another course . The process of getting the eigenvalues and eigenfunctions for this d ifferential operator is a subject for this course, and we illustrate h ow to derive the results here. " }}{PARA 221 "" 0 "" {TEXT -1 0 "" }} {PARA 225 "" 0 "" {TEXT 351 3 " " }}{PARA 226 "" 0 "" {TEXT 352 9 "Q uestion:" }{TEXT 352 1 " " }{TEXT 353 72 "What are all the eigenvalues of the selfadjoint, Sturm-Liouville Problem" }}{PARA 226 "" 0 "" {TEXT 353 10 " " }{TEXT 354 1 "y" }{TEXT 353 6 " '' = " } {XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT 353 1 " " }{TEXT 354 1 "y" } {TEXT 353 7 ", with " }{TEXT 354 1 "y" }{TEXT 353 6 "(0) = " }{TEXT 354 1 "y" }{TEXT 353 8 "(1) = 0?" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }} {PARA 226 "" 0 "" {TEXT 353 57 "We break the problem into two cases. F irst, suppose that " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT 353 41 " is positive. To remind us of this, take " }{XPPEDIT 18 0 "mu = lambda^2; " "6#/%#muG*$%'lambdaG\"\"#" }{TEXT 353 25 " . Thus, we seek numbers \+ " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 353 10 " such that" }} {PARA 226 "" 0 "" {TEXT 353 10 " " }{TEXT 354 1 "y" }{TEXT 353 6 " '' = " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }{TEXT 353 1 " " }{TEXT 354 1 "y" }{TEXT 353 6 " with " }{TEXT 354 1 "y" } {TEXT 353 6 "(0) = " }{TEXT 354 1 "y" }{TEXT 353 8 "(1) = 0." }}{PARA 226 "" 0 "" {TEXT 353 130 "The differential equation is a second order constant coefficient equation. We know how to solve it. Solutions are of the form exp(" }{TEXT 354 3 "r x" }{TEXT 353 53 "). In this case, \+ the general solutions is of the form" }}{PARA 226 "" 0 "" {TEXT 353 15 " " }{TEXT 354 1 "y" }{TEXT 353 1 "(" }{TEXT 354 1 "x " }{TEXT 353 11 ") = A exp( " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" } {TEXT 354 2 " x" }{TEXT 353 11 ") + B exp( " }{XPPEDIT 18 0 "-lambda; " "6#,$%'lambdaG!\"\"" }{TEXT 353 1 " " }{TEXT 354 1 "x" }{TEXT 353 2 ")." }}{PARA 226 "" 0 "" {TEXT 353 12 "To ask that " }{TEXT 354 1 "y" }{TEXT 353 41 "(0) = 0 asks that 0 = A + B. To ask that " }{TEXT 354 1 "y" }{TEXT 353 28 "(1) = 0 asks that 0 = A exp(" }{XPPEDIT 18 0 "lam bda;" "6#%'lambdaG" }{TEXT 353 10 ") + B/exp(" }{XPPEDIT 18 0 "lambda; " "6#%'lambdaG" }{TEXT 353 100 ") . The only solution is A = 0 = B. Th is is the trivial solutions and will not be of interest to us." }} {PARA 226 "" 0 "" {TEXT 353 29 " The second case is that " } {XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT 353 41 " is negative. To remind u s of this, take " }{XPPEDIT 18 0 "mu = -lambda^2;" "6#/%#muG,$*$%'lamb daG\"\"#!\"\"" }{TEXT 353 24 ". Thus, we seek numbers " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 353 11 " such that" }}{PARA 226 "" 0 "" {TEXT 353 10 " " }{TEXT 354 1 "y" }{TEXT 353 6 " '' = " }{XPPEDIT 18 0 "-lambda^2;" "6#,$*$%'lambdaG\"\"#!\"\"" }{TEXT 353 1 " " }{TEXT 354 1 "y" }{TEXT 353 6 " with " }{TEXT 354 1 "y" }{TEXT 353 6 "(0) = " }{TEXT 354 1 "y" }{TEXT 353 8 "(1) = 0." }}{PARA 225 "" 0 " " {TEXT 351 79 "We also know how to solve this differential equation. \+ Solutions are of the form" }}{PARA 226 "" 0 "" {TEXT 353 15 " \+ " }{TEXT 354 1 "y" }{TEXT 353 1 "(" }{TEXT 354 1 "x" }{TEXT 353 10 ") = A sin(" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 353 1 " \+ " }{TEXT 354 1 "x" }{TEXT 353 10 ") + B cos(" }{XPPEDIT 18 0 "lambda; " "6#%'lambdaG" }{TEXT 353 1 " " }{TEXT 354 1 "x" }{TEXT 353 2 ")." }} {PARA 226 "" 0 "" {TEXT 353 12 "To ask that " }{TEXT 354 1 "y" }{TEXT 353 42 "(0) = 0 asks that 0 = B. To also ask that " }{TEXT 354 1 "y" } {TEXT 353 28 "(1) = 0 asks that 0 = A sin(" }{XPPEDIT 18 0 "lambda;" " 6#%'lambdaG" }{TEXT 353 82 "). We don't want to have the trivial solut ions again, so it must be that 0 = sin(" }{XPPEDIT 18 0 "lambda;" "6# %'lambdaG" }{TEXT 353 81 ") . But we know all the places where the si ne function is zero. It must be that " }{XPPEDIT 18 0 "lambda = n*Pi; " "6#/%'lambdaG*&%\"nG\"\"\"%#PiGF'" }{TEXT 353 6 ", and " }{XPPEDIT 18 0 "mu = -n^2*Pi^2;" "6#/%#muG,$*&%\"nG\"\"#%#PiGF(!\"\"" }{TEXT 353 1 "." }}{PARA 226 "" 0 "" {TEXT 353 55 "The answer to the question is that the eigenvalues are " }{XPPEDIT 18 0 "-n^2*Pi^2;" "6#,$*&%\"n G\"\"#%#PiGF&!\"\"" }{TEXT 353 32 " and the eigenfuntions are sin(" } {XPPEDIT 18 0 "n*Pi*x;" "6#*(%\"nG\"\"\"%#PiGF%%\"xGF%" }{TEXT 353 2 " )." }}{PARA 225 "" 0 "" {TEXT 351 29 "Somehow, this is no surprise." } }{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 "" {TEXT 351 49 "We \+ do not forget the purpose of this development." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 352 7 "Fact 1:" }{TEXT 353 98 " Eigenvalues corresponding to self-adjoint transformations are real n umbers - not complex numbers." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }} {PARA 226 "" 0 "" {TEXT 352 7 "Fact 2:" }{TEXT 353 101 " Eigenvectors \+ for self adjoint transformations corresponding to different eigenvalue s are orthogonal." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 "" {TEXT 351 188 "Revisit Examples 3 and 4. Note that the eigenvectors are orthogonal. In the matrix example, this is seen quickly. For the \+ differential operator, recall that the dot product is an integral." }} {EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 18 "assume(m,integer):" } {MPLTEXT 1 350 37 "\nint(sin(n*Pi*x)*sin(m*Pi*x),x=0..1);" }}}{EXCHG {PARA 224 "" 0 "" {TEXT -1 0 "" }}}{PARA 221 "" 0 "" {TEXT -1 0 "" }} {PARA 226 "" 0 "" {TEXT 361 6 "Remark" }{TEXT 353 210 ". Concerning Fa ct 2 above, suppose we have two eigenvectors corresponding to one eige nvalue. The eigenvectors found may not be orthogonal. Not to worry! Pe rform the Gramm-Schmidt process to these eigenvectors. " }}{PARA 221 " " 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 "" {TEXT 351 178 "Most of the pa rtial differential equations that arise in this set of lecture notes w ill generate a special type of differential equation that is called a \+ Sturm-Liouville problem." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 352 48 "Definition of a regular Sturm-Liouville prob lem:" }{TEXT 353 14 " Suppose that " }{TEXT 354 1 "s" }{TEXT 353 2 ", \+ " }{TEXT 354 1 "s" }{TEXT 353 8 " ', and " }{TEXT 354 1 "q" }{TEXT 353 20 " are continuous and " }{XPPEDIT 18 0 "s(x) <> 0;" "6#0-%\"sG6# %\"xG\"\"!" }{TEXT 353 5 " for " }{TEXT 354 1 "x" }{TEXT 353 88 " in [ a,b]. In the presence of appropriate boundary conditions, the differen tial operator" }}{PARA 226 "" 0 "" {TEXT 353 13 " L(f) = (" } {TEXT 354 1 "s" }{TEXT 353 1 " " }{TEXT 354 1 "f" }{TEXT 353 8 " ') ' \+ - " }{TEXT 354 1 "q" }{TEXT 353 1 " " }{TEXT 354 1 "f" }}{PARA 226 "" 0 "" {TEXT 353 94 "is self adjoint in C ''([a,b]). Example of appropri ate boundary conditions are that functions " }{TEXT 354 1 "f" }{TEXT 353 38 " in the domain of L should satisfy are" }}{PARA 226 "" 0 "" {TEXT 353 4 "(1) " }{TEXT 354 1 "f" }{TEXT 353 5 "(0) =" }{TEXT 354 2 " f" }{TEXT 353 11 "(1) = 0, or" }}{PARA 226 "" 0 "" {TEXT 353 4 "(2) \+ " }{TEXT 354 1 "f" }{TEXT 353 8 " '(0) = " }{TEXT 354 1 "f" }{TEXT 353 13 " '(1) = 0, or" }}{PARA 226 "" 0 "" {TEXT 353 4 "(3) " }{TEXT 354 1 "f" }{TEXT 353 6 "(0) = " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" } {TEXT 353 1 " " }{TEXT 354 1 "f" }{TEXT 353 10 " '(0) and " }{TEXT 354 1 "f" }{TEXT 353 6 "(1) = " }{XPPEDIT 18 0 "-alpha;" "6#,$%&alphaG !\"\"" }{TEXT 353 1 " " }{TEXT 354 1 "f" }{TEXT 353 9 " '(1), or" }} {PARA 226 "" 0 "" {TEXT 353 7 "(4) if " }{TEXT 354 1 "s" }{TEXT 353 6 "(0) = " }{TEXT 354 1 "s" }{TEXT 353 10 "(1), then " }{TEXT 354 1 "f" }{TEXT 353 6 "(0) = " }{TEXT 354 1 "f" }{TEXT 353 8 "(1) and " }{TEXT 354 1 "f" }{TEXT 353 8 " '(0) = " }{TEXT 354 1 "f" }{TEXT 353 60 " '(1 ). (These last are called periodic boundary conditions.)" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 233 "" 0 "" {TEXT 362 9 "Example 7" }} {PARA 225 "" 0 "" {TEXT 351 95 "We provide two eigenvectors (eigenfunc tions) for each eigenvalue of the Sturm-Liouville problem" }}{PARA 226 "" 0 "" {TEXT 353 10 " " }{TEXT 354 1 "f" }{TEXT 353 6 " \+ '' = " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 353 1 " " }{TEXT 354 1 "f" }{TEXT 353 5 ", " }{TEXT 354 1 "f" }{TEXT 353 7 "(-1) = \+ " }{TEXT 354 1 "f" }{TEXT 353 6 "(1), " }{TEXT 354 1 "f" }{TEXT 353 9 " '(-1) = " }{TEXT 354 1 "f" }{TEXT 353 6 " '(1)." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 226 "" 0 "" {TEXT 353 148 "In this example, t here are an infinite number of eigenvalues, and for each one, there ar e two eigenfunctions. We will see that the eigenvalues are " } {XPPEDIT 18 0 "-n^2*pi^2;" "6#,$*&%\"nG\"\"#%#piGF&!\"\"" }{TEXT 353 22 " , for each integer n." }}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 18 "assume(n,integer);" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 19 "y1:=x->sin(n*Pi*x);" }{MPLTEXT 1 350 20 "\ny2:=x->cos(n*Pi*x); " }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 35 "A:=y->diff(y(x),x$2 )+n^2*Pi^2*y(x);" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 6 "A(y1 );" }{MPLTEXT 1 350 7 "\nA(y2);" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 14 "y1(-1); y1(1);" }{MPLTEXT 1 350 20 "\nD(y1)(-1);D(y 1)(1);" }}}{EXCHG {PARA 223 "> " 0 "" {MPLTEXT 1 350 14 "y2(-1); y2(1) ;" }{MPLTEXT 1 350 20 "\nD(y2)(-1);D(y2)(1);" }}}{EXCHG {PARA 224 "" 0 "" {TEXT -1 0 "" }}}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 "" {TEXT 351 93 "For later reference, we give the eigenvalues and ei genfunctions for the differential equation" }}{PARA 226 "" 0 "" {TEXT 353 18 " " }{TEXT 354 1 "y" }{TEXT 353 6 " '' = " } {XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT 354 2 " y" }}{PARA 225 " " 0 "" {TEXT 351 91 "for a variety of boundary conditions. Other bound ary conditions not mentioned are possible." }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 234 "" 0 "" {TEXT 363 30 "Eigenvalues and Eigenfunctio ns" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 235 "" 0 "" {TEXT 364 34 "Dirichlet Zero Boundary Conditions" }}{PARA 225 "" 0 "" {TEXT 351 35 "Boundary condition: y(0) = y(L) = 0" }}{PARA 226 "" 0 " " {TEXT 353 13 "Eigenvalues: " }{XPPEDIT 18 0 "-n^2*Pi^2/(L^2);" "6#,$ *(%\"nG\"\"#%#PiGF&*$%\"LGF&!\"\"F*" }{TEXT -1 0 "" }{TEXT 353 15 ", n = 1, 2, ..." }}{PARA 226 "" 0 "" {TEXT 353 21 "Eigenfunctions: sin(n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 353 20 "x/L) , n = 1, 2, ..." }}}{SECT 1 {PARA 236 "" 0 "" {TEXT 365 37 "Neumann Insulated Boundary \+ Conditions" }}{PARA 225 "" 0 "" {TEXT 351 37 "Boundary condition: y'(0 ) = y'(L) = 0" }}{PARA 226 "" 0 "" {TEXT 353 13 "Eigenvalues: " } {XPPEDIT 18 0 "-n^2*Pi^2/(L^2);" "6#,$*(%\"nG\"\"#%#PiGF&*$%\"LGF&!\" \"F*" }{TEXT 353 18 ", n = 0, 1, 2, ..." }}{PARA 226 "" 0 "" {TEXT 353 21 "Eigenfunctions: cos(n" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 353 23 "x/L) , n = 0, 1, 2, ..." }}}{SECT 1 {PARA 237 "" 0 "" {TEXT 366 28 "Periodic Boundary Conditions" }}{PARA 225 "" 0 "" {TEXT 351 46 "Boundary condition: y(0) = y(L), y'(0) = y'(L)" }}{PARA 226 "" 0 " " {TEXT 353 13 "Eigenvalues: " }{XPPEDIT 18 0 "-n^2*Pi^2/(L^2);" "6#,$ *(%\"nG\"\"#%#PiGF&*$%\"LGF&!\"\"F*" }{TEXT 353 17 ", n = 0,1, 2, ... " }}{PARA 226 "" 0 "" {TEXT 353 22 "Eigenfunctions: sin(n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 3 " x " }{TEXT 353 31 " /L) , n = 1, 2, ... and cos(n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 353 26 " x / L) , n = 0, 1, 2, ..." }}}{SECT 1 {PARA 238 "" 0 "" {TEXT 367 25 "Mixed \+ Boundary Conditions" }}{PARA 225 "" 0 "" {TEXT 351 37 "Boundary condit ion: y(0) = y'(L) = 0," }}{PARA 226 "" 0 "" {TEXT 353 14 "Eigenvalues: -" }{XPPEDIT 18 0 "`(`;" "6#%\"(G" }{TEXT -1 0 "" }{TEXT 353 1 "(" } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 353 7 "/2 + n " }{XPPEDIT 18 0 "P i;" "6#%#PiG" }{TEXT 353 3 ")/L" }{XPPEDIT 18 0 "`)`^2;" "6#*$%\")G\" \"#" }{TEXT -1 0 "" }{TEXT 353 18 ", n = 0, 1, 2, ..." }}{PARA 226 "" 0 "" {TEXT 353 21 "Eigenfunctions: sin((" }{XPPEDIT 18 0 "Pi;" "6#%#Pi G" }{TEXT 353 7 "/2 + n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 353 25 ") x /L), n = 0, 1, 2, ..." }}{PARA 225 "" 0 "" {TEXT 351 2 "OR" }} {PARA 225 "" 0 "" {TEXT 351 37 "Boundary condition: y'(0) = y(L) = 0, " }}{PARA 226 "" 0 "" {TEXT 353 15 "Eigenvalues: - " }{XPPEDIT 18 0 "` (`;" "6#%\"(G" }{TEXT -1 0 "" }{TEXT 353 1 "(" }{XPPEDIT 18 0 "Pi;" "6 #%#PiG" }{TEXT 353 7 "/2 + n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 353 4 ")/L " }{XPPEDIT 18 0 "`)`^2;" "6#*$%\")G\"\"#" }{TEXT -1 0 "" } {TEXT 353 18 ", n = 0, 1, 2, ..." }}{PARA 226 "" 0 "" {TEXT 353 21 "Ei genfunctions: cos((" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 353 7 "/2 + n " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 353 25 ") x /L), n = 0, 1, \+ 2, ..." }}}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 225 "" 0 "" {TEXT 351 50 "EMAIL: herod@math.gatech.edu or jherod@tds.net" }}{PARA 225 "" 0 "" {TEXT 351 38 "URL: http://www.math.gatech.edu/~herod" }} {PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 239 "" 0 "" {TEXT 368 36 "Copy right \251 2003 by James V. Herod" }}{PARA 239 "" 0 "" {TEXT 368 19 "All rights reserved" }}{PARA 221 "" 0 "" {TEXT -1 0 "" }}{PARA 240 " " 0 "" {TEXT -1 0 "" }}{PARA 241 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 42 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }