Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
And a few unproductive, but illustrative, examples.
sage: A = matrix(ZZ, 3, 4, range(12))
sage: B = column_matrix(ZZ, 3, 4, range(12))
sage: A == B.transpose()
True
sage: A = matrix(QQ, 7, 12, range(84))
sage: A == column_matrix(A.columns())
True
sage: A=column_matrix(QQ, matrix(ZZ, 3, 2, range(6)) )
sage: A
[0 2 4]
[1 3 5]
sage: A.parent()
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage.matrix.constructor.companion_matrix(poly, format=’right’)
This function is available as companion_matrix(...) and matrix.companion(...).
Create a companion matrix from a monic polynomial.
INPUT:
•poly - a univariate polynomial, or an iterable containing the coefficients of a polynomial, with low-degree
coefficients first. The polynomial (or the polynomial implied by the coefficients) must be monic. In other
words, the leading coefficient must be one. A symbolic expression that might also be a polynomial is not
proper input, see examples below.
•format - default: ‘right’ - specifies one of four variations of a companion matrix. Allowable values are
‘right’, ‘left’, ‘top’ and ‘bottom’, which indicates which border of the matrix contains the negatives of the
coefficients.
OUTPUT:
A square matrix with a size equal to the degree of the polynomial. The returned matrix has ones above, or below
the diagonal, and the negatives of the coefficients along the indicated border of the matrix (excepting the leading
one coefficient). See the first examples below for precise illustrations.
EXAMPLES:
Each of the four possibilities. Notice that the coefficients are specified and their negatives become the entries
of the matrix. The leading one must be given, but is not used. The permutation matrix P is the identity matrix,
with the columns reversed. The last three statements test the general relationships between the four variants.
sage: poly = [-2, -3, -4, -5, -6, 1]
sage: R = companion_matrix(poly, format=’right’); R
[0 0 0 0 2]
[1 0 0 0 3]
[0 1 0 0 4]
[0 0 1 0 5]
[0 0 0 1 6]
sage: L = companion_matrix(poly, format=’left’); L
[6 1 0 0 0]
[5 0 1 0 0]
[4 0 0 1 0]
[3 0 0 0 1]
[2 0 0 0 0]
sage: B = companion_matrix(poly, format=’bottom’); B
[0 1 0 0 0]
26 Chapter 2. Matrix Constructor