+
+ But if you supply a custom inner product, the result is
+ orthonormal with respect to that (and not the usual inner
+ product)::
+
+ sage: v1 = vector(QQ,(1,2,3))
+ sage: v2 = vector(QQ,(1,-1,6))
+ sage: v3 = vector(QQ,(2,1,-1))
+ sage: v = [v1,v2,v3]
+ sage: B = matrix(QQ, [ [6, 4, 2],
+ ....: [4, 5, 4],
+ ....: [2, 4, 9] ])
+ sage: ip = lambda x,y: (B*x).inner_product(y)
+ sage: norm = lambda x: ip(x,x)
+ sage: u = gram_schmidt(v,ip)
+ sage: all( norm(u_i) == 1 for u_i in u )
+ True
+ sage: ip(u[0],u[1]).is_zero()
+ True
+ sage: ip(u[0],u[2]).is_zero()
+ True
+ 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]
+ ]
+
+ It even works on Cartesian product spaces whose factors are vector
+ or matrix spaces::
+
+ sage: V1 = VectorSpace(AA,2)
+ sage: V2 = MatrixSpace(AA,2)
+ sage: M = cartesian_product([V1,V2])
+ sage: x1 = V1([1,1])
+ sage: x2 = V1([1,-1])
+ sage: y1 = V2.one()
+ sage: y2 = V2([0,1,1,0])
+ sage: z1 = M((x1,y1))
+ sage: z2 = M((x2,y2))
+ sage: def ip(a,b):
+ ....: return a[0].inner_product(b[0]) + (a[1]*b[1]).trace()
+ sage: U = gram_schmidt([z1,z2], inner_product=ip)
+ sage: ip(U[0],U[1])
+ 0
+ sage: ip(U[0],U[0])
+ 1
+ sage: ip(U[1],U[1])
+ 1
+