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 This computation is performed in a naive way using the ranks of powers of A−xI, where x is an eigenvalue of the matrix A. If desired, a transformation matrix P can be returned, which is such that the Jordan canonical form is given by P −1 AP . INPUT: •base_ring - Ring in which to compute the Jordan form. •sparse - (default False) If sparse=True, return a sparse matrix. •subdivide - (default True) If subdivide=True, the subdivisions for the Jordan blocks in the matrix are shown. •transformation - (default False) If transformation=True, computes also the transformation matrix. •eigenvalues - (default None) A complete set of roots, with multiplicity, of the characteristic polynomial of A, encoded as a list of pairs, each having the form (r, m) with r a root and m its multiplicity. If this is None, then Sage computes this list itself, but this is only possible over base rings in whose quotient fields polynomial factorization is implemented. Over all other rings, providing this list manually is the only way to compute Jordan normal forms. •check_input - (default True) A Boolean specifying whether the list eigenvalues (if provided) has to be checked for correctness. Set this to False for a speedup if the eigenvalues are known to be correct. NOTES: Currently, the Jordan normal form is not computed over inexact rings in any but the trivial cases when the matrix is either 0 × 0 or 1 × 1. In the case of exact rings, this method does not compute any generalized form of the Jordan normal form, but is only able to compute the result if the characteristic polynomial of the matrix splits over the specific base ring. Note that the base ring must be a field or a ring with an implemented fraction field. EXAMPLES: sage: a = matrix(ZZ,4,[1, 0, 0, 0, 0, 1, 0, 0, 1, \ -1, 1, 0, 1, -1, 1, 2]); a [ 1 0 0 0] [ 0 1 0 0] [ 1 -1 1 0] [ 1 -1 1 2] sage: a.jordan_form() [2|0 0|0] [-+---+-] [0|1 1|0] [0|0 1|0] [-+---+-] [0|0 0|1] sage: a.jordan_form(subdivide=False) [2 0 0 0] [0 1 1 0] [0 0 1 0] [0 0 0 1] sage: b = matrix(ZZ,3,range(9)); b [0 1 2] [3 4 5] [6 7 8] sage: b.jordan_form() 196 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 Traceback (most recent call last): ... RuntimeError: Some eigenvalue does not exist in Rational Field. sage: b.jordan_form(RealField(15)) Traceback (most recent call last): ... ValueError: Jordan normal form not implemented over inexact rings. If you need the transformation matrix as well as the Jordan form of self, then pass the option transformation=True. sage: m = matrix([[5,4,2,1],[0,1,-1,-1],[-1,-1,3,0],[1,1,-1,2]]); m [ 5 4 2 1] [ 0 1 -1 -1] [-1 -1 3 0] [ 1 1 -1 2] sage: jf, p = m.jordan_form(transformation=True) sage: jf [2|0|0 0] [-+-+---] [0|1|0 0] [-+-+---] [0|0|4 1] [0|0|0 4] sage: ~p * m * p [2 0 0 0] [0 1 0 0] [0 0 4 1] [0 0 0 4] Note that for matrices over inexact rings and associated numerical stability problems, we do not attempt to compute the Jordan normal form. sage: b = matrix(ZZ,3,3,range(9)) sage: jf, p = b.jordan_form(RealField(15), transformation=True) Traceback (most recent call last): ... ValueError: Jordan normal form not implemented over inexact rings. TESTS: sage: c = matrix(ZZ, 3, [1]*9); c [1 1 1] [1 1 1] [1 1 1] sage: c.jordan_form(subdivide=False) [3 0 0] [0 0 0] [0 0 0] sage: evals = [(i,i) for i in range(1,6)] sage: n = sum(range(1,6)) sage: jf = block_diagonal_matrix([jordan_block(ev,size) for ev,size in evals]) sage: p = random_matrix(ZZ,n,n) sage: while p.rank() != n: p = random_matrix(ZZ,n,n) sage: m = p * jf * ~p sage: mjf, mp = m.jordan_form(transformation=True) sage: mjf == jf True 197
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