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 sage: A = matrix(QQ, 2, 0) sage: A.left_kernel() Vector space of degree 2 and dimension 2 over Rational Field Basis matrix: [1 0] [0 1] sage: A = matrix(QQ, 0, 2) sage: A.left_kernel() Vector space of degree 0 and dimension 0 over Rational Field Basis matrix: [] The results are cached. Note that requesting a new format for the basis is ignored and the cached copy is returned. Work with a copy if you need a new left kernel, or perhaps investigate the right_kernel_matrix() method on the transpose, which does not cache its results and is more flexible. sage: A = matrix(QQ, [[1,1],[2,2]]) sage: K1 = A.left_kernel() sage: K1 Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [ 1 -1/2] sage: K2 = A.left_kernel() sage: K1 is K2 True sage: K3 = A.left_kernel(basis=’pivot’) sage: K3 Vector space of degree 2 and dimension 1 over Rational Field Basis matrix: [ 1 -1/2] sage: B = copy(A) sage: K3 = B.left_kernel(basis=’pivot’) sage: K3 Vector space of degree 2 and dimension 1 over Rational Field User basis matrix: [-2 1] sage: K3 is K1 False sage: K3 == K1 True left_nullity() Return the (left) nullity of this matrix, which is the dimension of the (left) kernel of this matrix acting from the right on row vectors. EXAMPLES: sage: M = Matrix(QQ,[[1,0,0,1],[0,1,1,0],[1,1,1,0]]) sage: M.nullity() 0 sage: M.left_nullity() 0 sage: A = M.transpose() sage: A.nullity() 1 sage: A.left_nullity() 1 212 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: M = M.change_ring(ZZ) sage: M.nullity() 0 sage: A = M.transpose() sage: A.nullity() 1 matrix_window(row=0, col=0, nrows=-1, ncols=-1, check=1) Return the requested matrix window. EXAMPLES: sage: A = matrix(QQ, 3, range(9)) sage: A.matrix_window(1,1, 2, 1) Matrix window of size 2 x 1 at (1,1): [0 1 2] [3 4 5] [6 7 8] We test the optional check flag. sage: matrix([1]).matrix_window(0,1,1,1, check=False) Matrix window of size 1 x 1 at (0,1): [1] sage: matrix([1]).matrix_window(0,1,1,1) Traceback (most recent call last): ... IndexError: matrix window index out of range Another test of bounds checking: sage: matrix([1]).matrix_window(1,1,1,1) Traceback (most recent call last): ... IndexError: matrix window index out of range maxspin(v) Computes the largest integer n such that the list of vectors S = [v, v∗A, ..., v∗A n ] are linearly independent, and returns that list. INPUT: •self - Matrix •v - Vector OUTPUT: •list - list of Vectors ALGORITHM: The current implementation just adds vectors to a vector space until the dimension doesn’t grow. This could be optimized by directly using matrices and doing an efficient Echelon form. Also, when the base is Q, maybe we could simultaneously keep track of what is going on in the reduction modulo p, which might make things much faster. EXAMPLES: sage: t = matrix(QQ, 3, range(9)); t [0 1 2] [3 4 5] [6 7 8] sage: v = (QQ^3).0 213
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:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 146 and 147:
Page 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
Page 166 and 167: Sage Reference Manual: Matrices and
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

Create successful ePaper yourself

Delete template?

Save as template?