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(x).arguments() (x,) sage: M(x+sin(x)).arguments() (x,) charpoly(var=’x’, algorithm=None) Compute the characteristic polynomial of self, using maxima. EXAMPLES: sage: M = matrix(SR, 2, 2, var(’a,b,c,d’)) sage: M.charpoly(’t’) t^2 + (-a - d)*t - b*c + a*d sage: matrix(SR, 5, [1..5^2]).charpoly() x^5 - 65*x^4 - 250*x^3 TESTS: The cached polynomial should be independent of the var argument (trac ticket #12292). We check (indirectly) that the second call uses the cached value by noting that its result is not cached: sage: M = MatrixSpace(SR, 2) sage: A = M(range(0, 2^2)) sage: type(A) sage: A.charpoly(’x’) x^2 - 3*x - 2 sage: A.charpoly(’y’) y^2 - 3*y - 2 sage: A._cache[’charpoly’] x^2 - 3*x - 2 eigenvalues(solution_set=False) Compute the eigenvalues by solving the characteristic polynomial in maxima EXAMPLES: sage: a=matrix(SR,[[1,2],[3,4]]) sage: a.eigenvalues() [-1/2*sqrt(33) + 5/2, 1/2*sqrt(33) + 5/2] eigenvectors_left() Compute the left eigenvectors of a matrix. For each distinct eigenvalue, returns a list of the form (e,V,n) where e is the eigenvalue, V is a list of eigenvectors forming a basis for the corresponding left eigenspace, and n is the algebraic multiplicity of the eigenvalue. EXAMPLES: sage: A = matrix(SR,3,3,range(9)); A [0 1 2] [3 4 5] [6 7 8] sage: es = A.eigenvectors_left(); es [(-3*sqrt(6) + 6, [(1, -1/5*sqrt(6) + 4/5, -2/5*sqrt(3)*sqrt(2) + 3/5)], 1), (3*sqrt(6) + 6, sage: eval, [evec], mult = es[0] sage: delta = eval*evec - evec*A sage: abs(abs(delta)) < 1e-10 abs(sqrt(1/25*(3*(2*sqrt(3)*sqrt(2) - 3)*(sqrt(6) - 2) + 16*sqrt(3)*sqrt(2) + 5*sqrt(6) - 54 306 Chapter 16. Symbolic matrices
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 sage: abs(abs(delta)).n() < 1e-10 True sage: A = matrix(SR, 2, 2, var(’a,b,c,d’)) sage: A.eigenvectors_left() [(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - sage: es = A.eigenvectors_left(); es [(1/2*a + 1/2*d - 1/2*sqrt(a^2 + 4*b*c - 2*a*d + d^2), [(1, -1/2*(a - d + sqrt(a^2 + 4*b*c - sage: eval, [evec], mult = es[0] sage: delta = eval*evec - evec*A sage: delta.apply_map(lambda x: x.full_simplify()) (0, 0) This routine calls Maxima and can struggle with even small matrices with a few variables, such as a 3 × 3 matrix with three variables. However, if the entries are integers or rationals it can produce exact values in a reasonable time. These examples create 0-1 matrices from the adjacency matrices of graphs and illustrate how the format and type of the results differ when the base ring changes. First for matrices over the rational numbers, then the same matrix but viewed as a symbolic matrix. sage: G=graphs.CycleGraph(5) sage: am = G.adjacency_matrix() sage: spectrum = am.eigenvectors_left() sage: qqbar_evalue = spectrum[2][0] sage: type(qqbar_evalue) sage: qqbar_evalue 0.618033988749895 sage: am = G.adjacency_matrix().change_ring(SR) sage: spectrum = am.eigenvectors_left() sage: symbolic_evalue = spectrum[2][0] sage: type(symbolic_evalue) sage: symbolic_evalue 1/2*sqrt(5) - 1/2 sage: qqbar_evalue == symbolic_evalue True A slightly larger matrix with a “nice” spectrum. sage: G=graphs.CycleGraph(6) sage: am = G.adjacency_matrix().change_ring(SR) sage: am.eigenvectors_left() [(-1, [(1, 0, -1, 1, 0, -1), (0, 1, -1, 0, 1, -1)], 2), (1, [(1, 0, -1, -1, 0, 1), (0, 1, 1, eigenvectors_right() Compute the right eigenvectors of a matrix. For each distinct eigenvalue, returns a list of the form (e,V,n) where e is the eigenvalue, V is a list of eigenvectors forming a basis for the corresponding right eigenspace, and n is the algebraic multiplicity of the eigenvalue. EXAMPLES: sage: A = matrix(SR,2,2,range(4)); A [0 1] [2 3] sage: right = A.eigenvectors_right(); right [(-1/2*sqrt(17) + 3/2, [(1, -1/2*sqrt(17) + 3/2)], 1), (1/2*sqrt(17) + 3/2, [(1, 1/2*sqrt(17 307
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Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

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