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 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.
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [ 3.0 4.0 5.0] sage: A.norm() doctest:...: DeprecationWarning: The default norm will be changing from p=’frob’ to p=2. See http://trac.sagemath.org/13643 for details. 8.30662386... sage: A.norm(p=’frob’) 8.30662386... sage: A.norm(p=Infinity) 12.0 sage: A.norm(p=-Infinity) 3.0 sage: A.norm(p=1) 8.0 sage: A.norm(p=-1) 6.0 sage: A.norm(p=2) 7.99575670... sage: A.norm(p=-2) < 10^-15 True Us And over the complex numbers. sage: B = matrix(CDF, 2, [[1+I, 2+3*I],[3+4*I,3*I]]); B [1.0 + 1.0*I 2.0 + 3.0*I] [3.0 + 4.0*I 3.0*I] sage: B.norm() 7.0 sage: B.norm(p=’frob’) 7.0 sage: B.norm(p=Infinity) 8.0 sage: B.norm(p=-Infinity) 5.01976483... sage: B.norm(p=1) 6.60555127... sage: B.norm(p=-1) 6.41421356... sage: B.norm(p=2) 6.66189877... sage: B.norm(p=-2) 2.14921023... Since it is invariant under unitary multiplication, the Frobenius norm is equal to the square root of the sum of squares of the singular values. sage: A = matrix(RDF, 5, range(1,26)) sage: f = A.norm(p=’frob’) sage: U, S, V = A.SVD() sage: s = sqrt(sum([S[i,i]^2 for i in range(5)])) sage: abs(f-s) < 1.0e-12 True Return values are in RDF . sage: A = matrix(CDF, 2, range(4)) sage: A.norm() in RDF True Improper values of p are caught. 379
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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