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 [ 1/4 -1/8 3 4 5] [ 0 1 2 -5/12 3/8] [ 3 4 5 1/4 -1/8] Without subdivisions it also deduces dimensions for scalars if possible: sage: C = matrix(ZZ, 1, 2, range(2)) sage: block_matrix([ [ C, 0 ], [ 3, 4 ], [ 5, 6, C ] ], subdivide=False ) [0 1 0 0] [3 0 4 0] [0 3 0 4] [5 6 0 1] If all submatrices are sparse (unless there are none at all), the result will be a sparse matrix. Otherwise it will be dense by default. The sparse keyword can be used to override this: sage: A = Matrix(ZZ, 2, 2, [0, 1, 0, 0], sparse=True) sage: block_matrix([ [ A ], [ A ] ]).parent() Full MatrixSpace of 4 by 2 sparse matrices over Integer Ring sage: block_matrix([ [ A ], [ A ] ], sparse=False).parent() Full MatrixSpace of 4 by 2 dense matrices over Integer Ring Consecutive zero submatrices are consolidated. sage: B = matrix(2, range(4)) sage: C = matrix(2, 8, range(16)) sage: block_matrix(2, [[B,0,0,B],[C]], subdivide=False) [ 0 1 0 0 0 0 0 1] [ 2 3 0 0 0 0 2 3] [ 0 1 2 3 4 5 6 7] [ 8 9 10 11 12 13 14 15] Ambiguity is not tolerated. sage: B = matrix(2, range(4)) sage: C = matrix(2, 8, range(16)) sage: block_matrix(2, [[B,0,B,0],[C]], subdivide=False) Traceback (most recent call last): ... ValueError: insufficient information to determine submatrix widths Historically, giving only a flat list of submatrices, whose number was a perfect square, would create a new matrix by laying out the submatrices in a square grid. This behavior is now deprecated. sage: A = matrix(2, 3, range(6)) sage: B = matrix(3, 3, range(9)) sage: block_matrix([A, A, B, B]) doctest:...: DeprecationWarning: invocation of block_matrix with just a list whose length is a p [0 1 2|0 1 2] [3 4 5|3 4 5] [-----+-----] [0 1 2|0 1 2] [3 4 5|3 4 5] [6 7 8|6 7 8] Historically, a flat list of matrices whose number is not a perfect square, with no specification of the number of rows or columns, would raise an error. This behavior continues, but could be removed when the deprecation above is completed. 24 Chapter 2. Matrix Constructor
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: A = matrix(2, 3, range(6)) sage: B = matrix(3, 3, range(9)) sage: block_matrix([A, A, A, B, B, B]) Traceback (most recent call last): ... ValueError: must specify nrows or ncols for non-square block matrix. sage.matrix.constructor.column_matrix(*args, **kwds) This function is available as column_matrix(...) and matrix.column(...). Constructs a matrix, and then swaps rows for columns and columns for rows. Note: Linear algebra in Sage favors rows over columns. So, generally, when creating a matrix, input vectors and lists are treated as rows. This function is a convenience that turns around this convention when creating a matrix. If you are not familiar with the usual matrix constructor, you might want to consider it first. INPUT: Inputs are almost exactly the same as for the matrix constructor, which are documented there. But see examples below for how dimensions are handled. OUTPUT: Output is exactly the transpose of what the matrix constructor would return. In other words, the matrix constructor builds a matrix and then this function exchanges rows for columns, and columns for rows. EXAMPLES: The most compelling use of this function is when you have a collection of lists or vectors that you would like to become the columns of a matrix. In almost any other situation, the matrix constructor can probably do the job just as easily, or easier. sage: col_1 = [1,2,3] sage: col_2 = [4,5,6] sage: column_matrix([col_1, col_2]) [1 4] [2 5] [3 6] sage: v1 = vector(QQ, [10, 20]) sage: v2 = vector(QQ, [30, 40]) sage: column_matrix(QQ, [v1, v2]) [10 30] [20 40] If you only specify one dimension along with a flat list of entries, then it will be the number of columns in the result (which is different from the behavior of the matrix constructor). sage: column_matrix(ZZ, 8, range(24)) [ 0 3 6 9 12 15 18 21] [ 1 4 7 10 13 16 19 22] [ 2 5 8 11 14 17 20 23] And when you specify two dimensions, then they should be number of columns first, then the number of rows, which is the reverse of how they would be specified for the matrix constructor. sage: column_matrix(QQ, 5, 3, range(15)) [ 0 3 6 9 12] [ 1 4 7 10 13] [ 2 5 8 11 14] 25
Page 1: Sage Reference Manual: Matrices and
Page 4 and 5: 22 Dense matrices over multivariate
Page 6 and 7: Sage Reference Manual: Matrices and
Page 78 and 79:
Sage Reference Manual: Matrices and
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:
Page 168 and 169:
Page 170 and 171:
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
Page 186 and 187:
Page 188 and 189:
Page 190 and 191:
Page 192 and 193:
Page 194 and 195:
Page 196 and 197:
Page 198 and 199:
Page 200 and 201:
Page 202 and 203:
Page 204 and 205:
Page 206 and 207:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
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?