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A=matrix(QQ,
[[  0,   -1,    0,    0,    1,    0,    0,    0,    0,    0,    0,    0],
[-1/2,    0,    0, -1/2,    0,    0,    1,    0,    0,    0,    0,    0],
[-1/2,   -1,    0,  1/2,    0,    0,    0,    1,    0,    0,    0,    0],
[-1/2,    0,    0, -1/2,    0,    0,    0,    0,    0,    1,    0,    0],
[ 1/2,   -1,    0, -1/2,    0,    0,    0,    0,    0,    0,    1,    0],
[   0,    0,   -1,    0,    0,   -1,    0,    0,    1,    0,    0,    1]])

G, M = A.gram_schmidt()     # no normalization in G
g=G*G.transpose()
gsqrt=g.apply_map(sqrt)
q=gsqrt.inverse()
Gr=q*G                      # Gr is the orthonormal version of G
print Gr*Gr.transpose()     # check the orthonormality of Gr
print (M*gsqrt)*Gr==A       # check if  A=(M*gsqrt)*Gr