Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: m.log_determinant()
0.69314718056
sage: m = matrix(RDF,0,0,[]); m
[]
sage: m.log_determinant()
0.0
sage: m = matrix(CDF,2,2,range(4)); m
[0.0 1.0]
[2.0 3.0]
sage: RDF(log(abs(m.determinant())))
0.69314718056
sage: m.log_determinant()
0.69314718056
sage: m = matrix(CDF,0,0,[]); m
[]
sage: m.log_determinant()
0.0
norm(p=None)
Returns the norm of the matrix.
INPUT:
•p - default: ‘frob’ - controls which norm is computed, allowable values are ‘frob’ (for the Frobenius
norm), integers -2, -1, 1, 2, positive and negative infinity. See output discussion for specifics. The
default (p=’frob’) is deprecated and will change to a default of p=2 soon.
OUTPUT:
Returned value is a double precision floating point value in RDF. Row and column sums described below
are sums of the absolute values of the entries, where the absolute value of the complex number a + bi is
√
a2 + b 2 . Singular values are the “diagonal” entries of the “S” matrix in the singular value decomposition.
•p = ’frob’: the Frobenius norm, which for a matrix A = (a ij ) computes
⎛
⎝ ∑ i,j
⎞1/2
|a i,j | 2 ⎠
•p = Infinity or p = oo: the maximum row sum.
•p = -Infinity or p = -oo: the minimum column sum.
•p = 1: the maximum column sum.
•p = -1: the minimum column sum.
•p = 2: the induced 2-norm, equal to the maximum singular value.
•p = -2: the minimum singular value.
ALGORITHM:
Computation is performed by the norm() function of the SciPy/NumPy library.
EXAMPLES:
First over the reals.
sage: A = matrix(RDF, 3, range(-3, 6)); A
[-3.0 -2.0 -1.0]
[ 0.0 1.0 2.0]
378 Chapter 19. Dense matrices using a NumPy backend.