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: eval, [evec], mult = es[0] sage: delta = eval*evec - evec*A sage: abs(abs(delta)) < 1e-10 True Notice the difference between considering ring extensions or not. sage: M=matrix(QQ,[[0,-1,0],[1,0,0],[0,0,2]]) sage: M.eigenvectors_left() [(2, [ (0, 0, 1) ], 1), (-1*I, [(1, -1*I, 0)], 1), (1*I, [(1, 1*I, 0)], 1)] sage: M.eigenvectors_left(extend=False) [(2, [ (0, 0, 1) ], 1)] eigenvectors_right(extend=True) 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. If extend = True (the default), this will return eigenspaces over the algebraic closure of the base field where this is implemented; otherwise it will restrict to eigenvalues in the base field. EXAMPLES: We compute the right eigenvectors 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: es = A.eigenvectors_right(); es [(0, [ (1, -2, 1) ], 1), (-1.348469228349535, [(1, 0.1303061543300932, -0.7393876913398137)], 1), (13.34846922834954, [(1, 3.069693845669907, 5.139387691339814)], 1)] sage: A.eigenvectors_right(extend=False) [(0, [ (1, -2, 1) ], 1)] sage: eval, [evec], mult = es[0] sage: delta = eval*evec - A*evec sage: abs(abs(delta)) < 1e-10 True elementary_divisors() If self is a matrix over a principal ideal domain R, return elements d i for 1 ≤ i ≤ k = min(r, s) where r and s are the number of rows and columns of self, such that the cokernel of self is isomorphic to R/(d 1 ) ⊕ R/(d 2 ) ⊕ R/(d k ) with d i | d i+1 for all i. These are the diagonal entries of the Smith form of self (see smith_form()). EXAMPLES: sage: OE = EquationOrder(x^2 - x + 2, ’w’) sage: w = OE.ring_generators()[0] sage: m = Matrix([ [1, w],[w,7]]) sage: m.elementary_divisors() [1, -w + 9] 166 Chapter 7. Base class for matrices, part 2
Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1 See Also: smith_form() elementwise_product(right) Returns the elementwise product of two matrices of the same size (also known as the Hadamard product). INPUT: •right - the right operand of the product. A matrix of the same size as self such that multiplication of elements of the base rings of self and right is defined, once Sage’s coercion model is applied. If the matrices have different sizes, or if multiplication of individual entries cannot be achieved, a TypeError will result. OUTPUT: A matrix of the same size as self and right. The entry in location (i, j) of the output is the product of the two entries in location (i, j) of self and right (in that order). The parent of the result is determined by Sage’s coercion model. If the base rings are identical, then the result is dense or sparse according to this property for the left operand. If the base rings must be adjusted for one, or both, matrices then the result will be sparse only if both operands are sparse. No subdivisions are present in the result. If the type of the result is not to your liking, or the ring could be “tighter,” adjust the operands with change_ring(). Adjust sparse versus dense inputs with the methods sparse_matrix() and dense_matrix(). EXAMPLES: sage: A = matrix(ZZ, 2, range(6)) sage: B = matrix(QQ, 2, [5, 1/3, 2/7, 11/2, -3/2, 8]) sage: C = A.elementwise_product(B) sage: C [ 0 1/3 4/7] [33/2 -6 40] sage: C.parent() Full MatrixSpace of 2 by 3 dense matrices over Rational Field Notice the base ring of the results in the next two examples. sage: D = matrix(ZZ[x],2,[1+x^2,2,3,4-x]) sage: E = matrix(QQ,2,[1,2,3,4]) sage: F = D.elementwise_product(E) sage: F [ x^2 + 1 4] [ 9 -4*x + 16] sage: F.parent() Full MatrixSpace of 2 by 2 dense matrices over Univariate Polynomial Ring in x over Rational sage: G = matrix(GF(3),2,[0,1,2,2]) sage: H = matrix(ZZ,2,[1,2,3,4]) sage: J = G.elementwise_product(H) sage: J [0 2] [0 2] sage: J.parent() Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 3 Non-commutative rings behave as expected. These are the usual quaternions. 167
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