Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

More documents

Recommendations

Info

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 NotImplementedError: eigenspaces cannot be computed reliably for inexact rings such as Real consult numerical or symbolic matrix classes for other options sage: em = B.change_ring(RDF).eigenmatrix_right() sage: eigenvalues = em[0]; eigenvalues.dense_matrix().zero_at(1e-15) [ 13.3484692283 0.0 0.0] [ 0.0 -1.34846922835 0.0] [ 0.0 0.0 0.0] sage: eigenvectors = em[1]; eigenvectors # not tested [ 0.164763817... 0.799699663... 0.408248290...] [ 0.505774475... 0.104205787... -0.816496580...] [ 0.846785134... -0.591288087... 0.408248290...] sage: x, y = var(’x y’) sage: S = matrix([[x, y], [y, 3*x^2]]) sage: em = S.eigenmatrix_right() sage: eigenvalues = em[0]; eigenvalues [3/2*x^2 + 1/2*x - 1/2*sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2) [ 0 3/2*x^2 + 1/2*x + 1/2*sqrt(9*x^4 - sage: eigenvectors = em[1]; eigenvectors [ 1 [1/2*(3*x^2 - x - sqrt(9*x^4 - 6*x^3 + x^2 + 4*y^2))/y 1/2*(3*x^2 - x + sqrt(9*x^4 - 6*x^3 + TESTS: sage: B = matrix(QQ, 2, 3, range(6)) sage: B.eigenspaces_right() Traceback (most recent call last): ... TypeError: matrix must be square, not 2 x 3 sage: B = matrix(QQ, 4, 4, range(16)) sage: B.eigenspaces_right(format=’junk’) Traceback (most recent call last): ... ValueError: format keyword must be None, ’all’ or ’galois’, not junk sage: B.eigenspaces_right(algebraic_multiplicity=’garbage’) Traceback (most recent call last): ... ValueError: algebraic_multiplicity keyword must be True or False eigenvalues(extend=True) Return a sequence of the eigenvalues of a matrix, with multiplicity. If the eigenvalues are roots of polynomials in QQ, then QQbar elements are returned that represent each separate root. If the option extend is set to False, only eigenvalues in the base ring are considered. EXAMPLES: sage: a = matrix(QQ, 4, range(16)); a [ 0 1 2 3] [ 4 5 6 7] [ 8 9 10 11] [12 13 14 15] sage: sorted(a.eigenvalues(), reverse=True) [32.46424919657298, 0, 0, -2.464249196572981] 164 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: a=matrix([(1, 9, -1, -1), (-2, 0, -10, 2), (-1, 0, 15, -2), (0, 1, 0, -1)]) sage: a.eigenvalues() [-0.9386318578049146, 15.50655435353258, 0.2160387521361705 - 4.713151979747493*I, 0.216 A symmetric matrix a+a.transpose() should have real eigenvalues sage: b=a+a.transpose() sage: ev = b.eigenvalues(); ev [-8.35066086057957, -1.107247901349379, 5.718651326708515, 33.73925743522043] The eigenvalues are elements of QQbar, so they really represent exact roots of polynomials, not just approximations. sage: e = ev[0]; e -8.35066086057957 sage: p = e.minpoly(); p x^4 - 30*x^3 - 171*x^2 + 1460*x + 1784 sage: p(e) == 0 True To perform computations on the eigenvalue as an element of a number field, you can always convert back to a number field element. sage: e.as_number_field_element() (Number Field in a with defining polynomial y^4 - 2*y^3 - 507*y^2 - 3972*y - 4264, a + 7, Ring morphism: From: Number Field in a with defining polynomial y^4 - 2*y^3 - 507*y^2 - 3972*y - 4264 To: Algebraic Real Field Defn: a |--> -15.35066086057957) Notice the effect of the extend option. sage: M=matrix(QQ,[[0,-1,0],[1,0,0],[0,0,2]]) sage: M.eigenvalues() [2, -1*I, 1*I] sage: M.eigenvalues(extend=False) [2] eigenvectors_left(extend=True) Compute the left eigenvectors of a matrix. For each distinct eigenvalue, returns a list of the form (e,V,n) where e is the eigenvalue, V is a list of eigenvectors forming a basis for the corresponding left eigenspace, and n is the algebraic multiplicity of the eigenvalue. If the option extend is set to False, then only the eigenvalues that live in the base ring are considered. EXAMPLES: We compute the left eigenvectors of a 3 × 3 rational matrix. sage: A = matrix(QQ,3,3,range(9)); A [0 1 2] [3 4 5] [6 7 8] sage: es = A.eigenvectors_left(); es [(0, [ (1, -2, 1) ], 1), (-1.348469228349535, [(1, 0.3101020514433644, -0.3797958971132713)], 1), (13.34846922834954, [(1, 1.289897948556636, 1.579795897113272)], 1)] 165
Page 1:
Sage Reference Manual: Matrices and
Page 4 and 5:
22 Dense matrices over multivariate
Page 6 and 7:
Page 8 and 9:
Page 10 and 11:
Page 12 and 13:
Page 14 and 15:
Page 16 and 17:
Page 18 and 19:
Page 20 and 21:
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119: Sage Reference Manual: Matrices and
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
Page 250 and 251:
Page 252 and 253:
Page 254 and 255:
Page 256 and 257:
Page 258 and 259:
Page 260 and 261:
Page 262 and 263:
Page 264 and 265:
Page 266 and 267:
Page 268 and 269:
Page 270 and 271:
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
Page 308 and 309:
Page 310 and 311:
Page 312 and 313:
Page 314 and 315:
Page 316 and 317:
Page 318 and 319:
Page 320 and 321:
Page 322 and 323:
Page 324 and 325:
Page 326 and 327:
Page 328 and 329:
Page 330 and 331:
Page 332 and 333:
Page 334 and 335:
Page 336 and 337:
Page 338 and 339:
Page 340 and 341:
Page 342 and 343:
Page 344 and 345:
Page 346 and 347:
Page 348 and 349:
Page 350 and 351:
Page 352 and 353:
Page 354 and 355:
Page 356 and 357:
Page 358 and 359:
Page 360 and 361:
Page 362 and 363:
Page 364 and 365:
Page 366 and 367:
Page 368 and 369:
Page 370 and 371:
Page 372 and 373:
Page 374 and 375:
Page 376 and 377:
Page 378 and 379:
Page 380 and 381:
Page 382 and 383:
Page 384 and 385:
Page 386 and 387:
Page 388 and 389:
Page 390 and 391:
Page 392 and 393:
Page 394 and 395:
Page 396 and 397:
Page 398 and 399:
Page 400 and 401:
Page 402 and 403:
Page 404 and 405:
Page 406 and 407:
Page 408 and 409:
Page 410 and 411:
Page 412 and 413:
Page 414 and 415:
Page 416 and 417:
Page 418 and 419:
Page 420 and 421:
Page 422 and 423:
Page 424 and 425:
Page 426 and 427:
Page 428 and 429:
Page 430 and 431:
Page 432 and 433:
Page 434 and 435:
Page 436 and 437:
Page 438 and 439:
Page 440 and 441:
Page 442 and 443:
show all

Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?