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 ... ValueError: format keyword must be None, ’all’ or ’galois’, not junk sage: B.eigenspaces_left(algebraic_multiplicity=’garbage’) Traceback (most recent call last): ... ValueError: algebraic_multiplicity keyword must be True or False eigenspaces_right(format=’all’, var=’a’, algebraic_multiplicity=False) Compute the right eigenspaces of a matrix. Note that eigenspaces_right() and right_eigenspaces() are identical methods. “right” refers to the eigenvectors being placed to the right of the matrix. INPUT: •self - a square matrix over an exact field. For inexact matrices consult the numerical or symbolic matrix classes. •format - default: None –’all’ - attempts to create every eigenspace. This will always be possible for matrices with rational entries. –’galois’ - for each irreducible factor of the characteristic polynomial, a single eigenspace will be output for a single root/eigenvalue for the irreducible factor. –None - Uses the ‘all’ format if the base ring is contained in an algebraically closed field which is implemented. Otherwise, uses the ‘galois’ format. •var - default: ‘a’ - variable name used to represent elements of the root field of each irreducible factor of the characteristic polynomial. If var=’a’, then the root fields will be in terms of a0, a1, a2, ...., where the numbering runs across all the irreducible factors of the characteristic polynomial, even for linear factors. •algebraic_multiplicity - default: False - whether or not to include the algebraic multiplicity of each eigenvalue in the output. See the discussion below. OUTPUT: If algebraic_multiplicity=False, return a list of pairs (e, V) where e is an eigenvalue of the matrix, and V is the corresponding left eigenspace. For Galois conjugates of eigenvalues, there may be just one representative eigenspace, depending on the format keyword. If algebraic_multiplicity=True, return a list of triples (e, V, n) where e and V are as above and n is the algebraic multiplicity of the eigenvalue. Warning: Uses a somewhat naive algorithm (simply factors the characteristic polynomial and computes kernels directly over the extension field). EXAMPLES: Right eigenspaces are computed from the left eigenspaces of the transpose of the matrix. As such, there is a greater collection of illustrative examples at the eigenspaces_left(). We compute the right eigenspaces of a 3 × 3 rational matrix. sage: A = matrix(QQ, 3 ,3, range(9)); A [0 1 2] [3 4 5] [6 7 8] sage: A.eigenspaces_right() Here 162 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 [ (0, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -2 1]), (-1.348469228349535, Vector space of degree 3 and dimension 1 over Algebraic Field User basis matrix: [ 1 0.1303061543300932 -0.7393876913398137]), (13.34846922834954, Vector space of degree 3 and dimension 1 over Algebraic Field User basis matrix: [ 1 3.069693845669907 5.139387691339814]) ] sage: es = A.eigenspaces_right(format=’galois’); es [ (0, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -2 1]), (a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomi User basis matrix: [ 1 1/5*a1 + 2/5 2/5*a1 - 1/5]) ] sage: es = A.eigenspaces_right(format=’galois’, algebraic_multiplicity=True); es [ (0, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -2 1], 1), (a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomi User basis matrix: [ 1 1/5*a1 + 2/5 2/5*a1 - 1/5], 1) ] sage: e, v, n = es[0]; v = v.basis()[0] sage: delta = v*e - A*v sage: abs(abs(delta)) < 1e-10 True The same computation, but with implicit base change to a field: sage: A = matrix(ZZ, 3, range(9)); A [0 1 2] [3 4 5] [6 7 8] sage: A.eigenspaces_right(format=’galois’) [ (0, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [ 1 -2 1]), (a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomi User basis matrix: [ 1 1/5*a1 + 2/5 2/5*a1 - 1/5]) ] This method is only applicable to exact matrices. The “eigenmatrix” routines for matrices with doubleprecision floating-point entries (RDF, CDF) are the best alternative. (Since some platforms return eigenvectors that are the negatives of those given here, this one example is not tested here.) There are also “eigenmatrix” routines for matrices with symbolic entries. sage: B = matrix(RR, 3, 3, range(9)) sage: B.eigenspaces_right() Traceback (most recent call last): ... 163
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