Sage Reference Manual: Matrices and Spaces of Matrices

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.2A negative size sometimes causes the error that there are too many elements.sage: diagonal_matrix(-2, [2])Traceback (most recent call last):...ValueError: number of diagonal matrix entries (1) exceeds the requested matrix size (-2)Types for the entries are limited, even though they may have a length.sage: diagonal_matrix(x^2)Traceback (most recent call last):...TypeError: diagonal matrix entries are not a supported type (list, tuple, vector, or NumPy arrayAUTHOR:•Rob Beezer (2011-01-11): total rewritesage.matrix.constructor.elementary_matrix(arg0, arg1=None, **kwds)This function is available as elementary_matrix(...) and matrix.elementary(...).Creates a square matrix that corresponds to a row operation or a column operation.FORMATS:In each case, R is the base ring, and is optional. n is the size of the square matrix created. Any call mayinclude the sparse keyword to determine the representation used. The default is False which leads to adense representation. We describe the matrices by their associated row operation, see the output description formore.•elementary_matrix(R, n, row1=i, row2=j)The matrix which swaps rows i and j.•elementary_matrix(R, n, row1=i, scale=s)The matrix which multiplies row i by s.•elementary_matrix(R, n, row1=i, row2=j, scale=s)The matrix which multiplies row j by s and adds it to row i.Elementary matrices representing column operations are created in an entirely analogous way, replacing row1by col1 and replacing row2 by col2.Specifying the ring for entries of the matrix is optional. If it is not given, and a scale parameter is provided, thena ring containing the value of scale will be used. Otherwise, the ring defaults to the integers.OUTPUT:An elementary matrix is a square matrix that is very close to being an identity matrix. If E is an elementarymatrix and A is any matrix with the same number of rows, then E*A is the result of applying a row operationto A. This is how the three types created by this function are described. Similarly, an elementary matrix canbe associated with a column operation, so if E has the same number of columns as A then A*E is the result ofperforming a column operation on A.An elementary matrix representing a row operation is created if row1 is specified, while an elementary matrixrepresenting a column operation is created if col1 is specified.EXAMPLES:Over the integers, creating row operations. Recall that row and column numbering begins at zero.31