Sage Reference Manual: Matrices and Spaces of Matrices

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.2[12 15 14 13][16 19 18 17]sage: E = elementary_matrix(QQ, 4, col1=2, scale=1/2); E[ 1 0 0 0][ 0 1 0 0][ 0 0 1/2 0][ 0 0 0 1]sage: A*E[ 0 1 1 3][ 4 5 3 7][ 8 9 5 11][12 13 7 15][16 17 9 19]sage: E = elementary_matrix(QQ, 4, col1=2, col2=1, scale=10); E[ 1 0 0 0][ 0 1 10 0][ 0 0 1 0][ 0 0 0 1]sage: A*E[ 0 1 12 3][ 4 5 56 7][ 8 9 100 11][ 12 13 144 15][ 16 17 188 19]An elementary matrix is always nonsingular. Then repeated row operations can be represented by productsof elementary matrices, and this product is again nonsingular. If row operations are to preserve fundamentalproperties of a matrix (like rank), we do not allow scaling a row by zero. Similarly, the corresponding elementarymatrix is not constructed. Also, we do not allow adding a multiple of a row to itself, since this could also leadto a new zero row.sage: A = matrix(QQ, 4, 10, range(40)); A[ 0 1 2 3 4 5 6 7 8 9][10 11 12 13 14 15 16 17 18 19][20 21 22 23 24 25 26 27 28 29][30 31 32 33 34 35 36 37 38 39]sage: E1 = elementary_matrix(QQ, 4, row1=0, row2=1)sage: E2 = elementary_matrix(QQ, 4, row1=3, row2=0, scale=100)sage: E = E2*E1sage: E.is_singular()Falsesage: E*A[ 10 11 12 13 14 15 16 17 18 19][ 0 1 2 3 4 5 6 7 8 9][ 20 21 22 23 24 25 26 27 28 29][1030 1131 1232 1333 1434 1535 1636 1737 1838 1939]sage: E3 = elementary_matrix(QQ, 4, row1=3, scale=0)Traceback (most recent call last):...ValueError: scale parameter of row of elementary matrix must be non-zerosage: E4 = elementary_matrix(QQ, 4, row1=3, row2=3, scale=12)Traceback (most recent call last):...33