code: def solve_sylvester(a, b, q):
      if (
          a.shape == b.shape
          and _torch.all(a == b)
          and _torch.all(_torch.abs(a - a.transpose(-2, -1)) < 1e-6)
      ):
          eigvals, eigvecs = eigh(a)
          if _torch.all(eigvals >= 1e-6):
              tilde_q = eigvecs.transpose(-2, -1) @ q @ eigvecs
              tilde_x = tilde_q / (
                  eigvals[..., :, None] + eigvals[..., None, :]
              )
              return eigvecs @ tilde_x @ eigvecs.transpose(-2, -1)
  
          conditions = _torch.all(eigvals >= 1e-6) or (
              a.shape[-1] >= 2.0
              and _torch.all(eigvals[..., 0] > -1e-6)
              and _torch.all(eigvals[..., 1] >= 1e-6)
              and _torch.all(_torch.abs(q + q.transpose(-2, -1)) < 1e-6)
          )
          if conditions:
              tilde_q = eigvecs.transpose(-2, -1) @ q @ eigvecs
              tilde_x = tilde_q / (
                  eigvals[..., :, None]
                  + eigvals[..., None, :]
                  + _torch.eye(a.shape[-1])
              )
              return eigvecs @ tilde_x @ eigvecs.transpose(-2, -1)
  
      solution = _np.vectorize(
          _scipy.linalg.solve_sylvester,
          signature="(m,m),(n,n),(m,n)->(m,n)",
      )(a, b, q)
      return _torch.from_numpy(solution)
  
file_path: geomstats/geomstats/_backend/pytorch/linalg.py
imports: import numpy
  import scipy
  import torch
  from .._backend_config import np_atol
  from ..numpy import linalg
  from ._dtype import _cast_out_to_input_dtype
label_desc: Use image node_id 6 for calling a global function with example usage: solve_sylvester(a, b, q) and returns: _torch, unknown, unknown