From 6d875d564f6d962f855c520c21cf539bdd4a3c1d Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 7 Dec 2020 15:48:19 -0500 Subject: [PATCH] eja: add example of Gram-Schmidt with matrices. --- mjo/eja/eja_utils.py | 20 +++++++++++++++++--- 1 file changed, 17 insertions(+), 3 deletions(-) diff --git a/mjo/eja/eja_utils.py b/mjo/eja/eja_utils.py index 4d70e06..b6e0c7d 100644 --- a/mjo/eja/eja_utils.py +++ b/mjo/eja/eja_utils.py @@ -1,9 +1,6 @@ from sage.functions.other import sqrt from sage.matrix.constructor import matrix from sage.modules.free_module_element import vector -from sage.rings.number_field.number_field import NumberField -from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing -from sage.rings.real_lazy import RLF def _mat2vec(m): return vector(m.base_ring(), m.list()) @@ -64,6 +61,23 @@ def gram_schmidt(v, inner_product=None): sage: ip(u[1],u[2]).is_zero() True + This Gram-Schmidt routine can be used on matrices as well, so long + as an appropriate inner-product is provided:: + + sage: E11 = matrix(QQ, [ [1,0], + ....: [0,0] ]) + sage: E12 = matrix(QQ, [ [0,1], + ....: [1,0] ]) + sage: E22 = matrix(QQ, [ [0,0], + ....: [0,1] ]) + sage: I = matrix.identity(QQ,2) + sage: trace_ip = lambda X,Y: (X*Y).trace() + sage: gram_schmidt([E11,E12,I,E22], inner_product=trace_ip) + [ + [1 0] [ 0 1/2*sqrt(2)] [0 0] + [0 0], [1/2*sqrt(2) 0], [0 1] + ] + TESTS: Ensure that zero vectors don't get in the way:: -- 2.43.2