Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 OUTPUT: A nonnegative integer. EXAMPLE: sage: a = Mat(ZZ,3)(range(9)) sage: a.height() 8 sage: a = Mat(ZZ,2,3)([-17,3,-389,15,-1,0]); a [ -17 3 -389] [ 15 -1 0] sage: a.height() 389 hermite_form(algorithm=’default’, proof=None, include_zero_rows=True, transformation=False, D=None) Return the echelon form of this matrix over the integers, also known as the hermit normal form (HNF). INPUT: •algorithm – String. The algorithm to use. Valid options are: –’default’ – Let Sage pick an algorithm (default). Up to 10 rows or columns: pari with flag 0; Up to 75 rows or columns: pari with flag 1; Larger: use padic algorithm. –’padic’ - an asymptotically fast p-adic modular algorithm, If your matrix has large coefficients and is small, you may also want to try this. –’pari’ - use PARI with flag 1 –’pari0’ - use PARI with flag 0 –’pari4’ - use PARI with flag 4 (use heuristic LLL) –’ntl’ - use NTL (only works for square matrices of full rank!) •proof - (default: True); if proof=False certain determinants are computed using a randomized hybrid p-adic multimodular strategy until it stabilizes twice (instead of up to the Hadamard bound). It is incredibly unlikely that one would ever get an incorrect result with proof=False. •include_zero_rows - (default: True) if False, don’t include zero rows •transformation - if given, also compute transformation matrix; only valid for padic algorithm •D - (default: None) if given and the algorithm is ‘ntl’, then D must be a multiple of the determinant and this function will use that fact. OUTPUT: The Hermite normal form (=echelon form over Z) of self. EXAMPLES: sage: A = MatrixSpace(ZZ,2)([1,2,3,4]) sage: A.echelon_form() [1 0] [0 2] sage: A = MatrixSpace(ZZ,5)(range(25)) sage: A.echelon_form() [ 5 0 -5 -10 -15] [ 0 1 2 3 4] [ 0 0 0 0 0] [ 0 0 0 0 0] [ 0 0 0 0 0] Getting a transformation matrix in the nonsquare case: 326 Chapter 17. Dense matrices over the integer ring
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: A = matrix(ZZ,5,3,[1..15]) sage: H, U = A.hermite_form(transformation=True, include_zero_rows=False) sage: H [1 2 3] [0 3 6] sage: U [ 0 0 0 4 -3] [ 0 0 0 13 -10] sage: U*A == H True TESTS: Make sure the zero matrices are handled correctly: sage: m = matrix(ZZ,3,3,[0]*9) sage: m.echelon_form() [0 0 0] [0 0 0] [0 0 0] sage: m = matrix(ZZ,3,1,[0]*3) sage: m.echelon_form() [0] [0] [0] sage: m = matrix(ZZ,1,3,[0]*3) sage: m.echelon_form() [0 0 0] The ultimate border case! sage: m = matrix(ZZ,0,0,[]) sage: m.echelon_form() [] Note: If ‘ntl’ is chosen for a non square matrix this function raises a ValueError. Special cases: 0 or 1 rows: sage: a = matrix(ZZ, 1,2,[0,-1]) sage: a.hermite_form() [0 1] sage: a.pivots() (1,) sage: a = matrix(ZZ, 1,2,[0,0]) sage: a.hermite_form() [0 0] sage: a.pivots() () sage: a = matrix(ZZ,1,3); a [0 0 0] sage: a.echelon_form(include_zero_rows=False) [] sage: a.echelon_form(include_zero_rows=True) [0 0 0] Illustrate using various algorithms.: sage: matrix(ZZ,3,[1..9]).hermite_form(algorithm=’pari’) [1 2 3] 327
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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