It seems that the symeig() 1 function does not return proper eigenvectors.  See the following simple test script:
import torch

# Create a random symmetric matrix
p, q = 10, 3
torch.manual_seed(0)
in_tensor = torch.rand(p, q, dtype=torch.float64, requires_grad=True).cuda()
cov_in = torch.mm(in_tensor.t(), in_tensor)

_, eig_vecs = torch.symeig(cov_in)

print(eig_vecs)
print(torch.mm(eig_vecs, eig_vecs.t()))
print(torch.mm(eig_vecs.t(), eig_vecs))
print(eig_vecs.norm(dim=0))
print(eig_vecs.norm(dim=1))

Here is the result:
tensor([[ 0.3573,  0.0288,  0.4334],
        [ 3.1050,  6.5767, -3.2518],
        [ 2.3730,  2.2232,  2.1961]],
       device='cuda:0', dtype=torch.float64, grad_fn=<SymeigBackward>)
tensor([[ 0.3164, -0.1103,  1.8639],
        [-0.1103, 63.4675, 14.8483],
        [ 1.8639, 14.8483, 15.3969]],
       device='cuda:0', dtype=torch.float64, grad_fn=<MmBackward>)
tensor([[ 15.3995,  25.7064,  -4.7303],
        [ 25.7064,  48.1965, -16.4908],
        [ -4.7303, -16.4908,  15.5848]],
       device='cuda:0', dtype=torch.float64, grad_fn=<MmBackward>)
tensor([3.9242, 6.9424, 3.9478],
       device='cuda:0', dtype=torch.float64, grad_fn=<NormBackward1>)
tensor([0.5625, 7.9667, 3.9239],
       device='cuda:0', dtype=torch.float64, grad_fn=<NormBackward1>)

Is there a bug in the symeig() function?