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 [ 7 - 3*I -1 - 6*I 3 - 5*I] [ 3 - 3*I -3 - 6*I 5 - 1*I] H Returns the conjugate-transpose (Hermitian) matrix. EXAMPLE: sage: A = matrix(QQbar, [[ -3, 5 - 3*I, 7 - 4*I], ... [7 + 3*I, -1 + 6*I, 3 + 5*I], ... [3 + 3*I, -3 + 6*I, 5 + I]]) sage: A.H [ -3 7 - 3*I 3 - 3*I] [ 5 + 3*I -1 - 6*I -3 - 6*I] [ 7 + 4*I 3 - 5*I 5 - 1*I] I Returns the inverse of the matrix, if it exists. EXAMPLES: sage: A = matrix(QQ, [[-5, -3, -1, -7], ... [ 1, 1, 1, 0], ... [-1, -2, -2, 0], ... [-2, -1, 0, -4]]) sage: A.I [ 0 2 1 0] [-4 -8 -2 7] [ 4 7 1 -7] [ 1 1 0 -2] sage: B = matrix(QQ, [[-11, -5, 18, -6], ... [ 1, 2, -6, 8], ... [ -4, -2, 7, -3], ... [ 1, -2, 5, -11]]) sage: B.I Traceback (most recent call last): ... ZeroDivisionError: input matrix must be nonsingular LU(pivot=None, format=’plu’) Finds a decomposition into a lower-triangular matrix and an upper-triangular matrix. INPUT: •pivot - pivoting strategy –‘auto’ (default) - see if the matrix entries are ordered (i.e. if they have an absolute value method), and if so, use a the partial pivoting strategy. Otherwise, fall back to the nonzero strategy. This is the best choice for general routines that may call this for matrix entries of a variety of types. –‘partial’ - each column is examined for the element with the largest absolute value and the row containing this element is swapped into place. –‘nonzero’ - the first nonzero element in a column is located and the row with this element is used. •format - contents of output, see more discussion below about output. –‘plu’ (default) - a triple; matrices P, L and U such that A = P*L*U. –‘compact’ - a pair; row permutation as a tuple, and the matrices L and U combined into one matrix. 120 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 OUTPUT: Suppose that A is an m × n matrix, then an LU decomposition is a lower-triangular m × m matrix L with every diagonal element equal to 1, and an upper-triangular m × n matrix, U such that the product LU, after a permutation of the rows, is then equal to A. For the ‘plu’ format the permutation is returned as an m × m permutation matrix P such that A = P LU It is more common to place the permutation matrix just to the left of A. If you desire this version, then use the inverse of P which is computed most efficiently as its transpose. If the ‘partial’ pivoting strategy is used, then the non-diagonal entries of L will be less than or equal to 1 in absolute value. The ‘nonzero’ pivot strategy may be faster, but the growth of data structures for elements of the decomposition might counteract the advantage. By necessity, returned matrices have a base ring equal to the fraction field of the base ring of the original matrix. In the ‘compact’ format, the first returned value is a tuple that is a permutation of the rows of LU that yields A. See the doctest for how you might employ this permutation. Then the matrices L and U are merged into one matrix – remove the diagonal of ones in L and the remaining nonzero entries can replace the entries of U beneath the diagonal. The results are cached, only in the compact format, separately for each pivot strategy called. Repeated requests for the ‘plu’ format will require just a small amount of overhead in each call to bust out the compact format to the three matrices. Since only the compact format is cached, the components of the compact format are immutable, while the components of the ‘plu’ format are regenerated, and hence are mutable. Notice that while U is similar to row-echelon form and the rows of U span the row space of A, the rows of U are not generally linearly independent. Nor are the pivot columns (or rank) immediately obvious. However for rings without specialized echelon form routines, this method is about twice as fast as the generic echelon form routine since it only acts “below the diagonal”, as would be predicted from a theoretical analysis of the algorithms. Note: This is an exact computation, so limited to exact rings. If you need numerical results, convert the base ring to the field of real double numbers, RDF or the field of complex double numbers, CDF, which will use a faster routine that is careful about numerical subtleties. ALGORITHM: “Gaussian Elimination with Partial Pivoting,” Algorithm 21.1 of [TREFETHEN-BAU]. EXAMPLES: Notice the difference in the L matrix as a result of different pivoting strategies. With partial pivoting, every entry of L has absolute value 1 or less. sage: A = matrix(QQ, [[1, -1, 0, 2, 4, 7, -1], ... [2, -1, 0, 6, 4, 8, -2], ... [2, 0, 1, 4, 2, 6, 0], ... [1, 0, -1, 8, -1, -1, -3], ... [1, 1, 2, -2, -1, 1, 3]]) sage: P, L, U = A.LU(pivot=’partial’) sage: P [0 0 0 0 1] [1 0 0 0 0] [0 0 0 1 0] 121
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