code: def solve_det_marginal(x, sigma, A, Q):
      """
      Use matrix inversion lemma for the solve:
  
      .. math::
          (\\Sigma + AQA^T)^{-1} X =\\
          (\\Sigma^{-1} - \\Sigma^{-1} A (Q^{-1} +
          A^T \\Sigma^{-1} A)^{-1} A^T \\Sigma^{-1}) X
  
      Use matrix determinant lemma for determinant:
  
      .. math::
          \\log|(\\Sigma + AQA^T)| = \\log|Q^{-1} + A^T \\Sigma^{-1} A|
          + \\log|Q| + \\log|\\Sigma|
  
      Parameters
      ----------
      x: tf.Tensor
          Tensor to multiply the solve by
      sigma: brainiak.matnormal.CovBase
          Covariance object implementing solve and logdet
      A: tf.Tensor
          Factor multiplying the variable we marginalized out
      Q: brainiak.matnormal.CovBase
          Covariance object of marginalized variable,
          implementing solve and logdet
      """
  
      # For diagnostics, we want to check condition numbers
      # of things we invert. This includes Q and Sigma, as well
      # as the "lemma factor" for lack of a better definition
      logging.log(
          logging.DEBUG, "Printing diagnostics for solve_det_marginal"
      )
      lemma_cond = _condition(
          Q._prec + tf.matmul(A, sigma.solve(A), transpose_a=True)
      )
      logging.log(
          logging.DEBUG,
          f"lemma_factor condition={lemma_cond}",
      )
      logging.log(logging.DEBUG, f"Q condition={_condition(Q._cov)}")
      logging.log(
          logging.DEBUG, f"sigma condition={_condition(sigma._cov)}"
      )
      logging.log(
          logging.DEBUG,
          f"sigma max={tf.reduce_max(input_tensor=A)},"
          + f"sigma min={tf.reduce_min(input_tensor=A)}",
      )
  
      # cholesky of (Qinv + A^T Sigma^{-1} A), which looks sort of like
      # a schur complement but isn't, so we call it the "lemma factor"
      # since we use it in woodbury and matrix determinant lemmas
      lemma_factor = tlinalg.cholesky(
          Q._prec + tf.matmul(A, sigma.solve(A), transpose_a=True)
      )
  
      logdet = (
          Q.logdet
          + sigma.logdet
          + 2
          * tf.reduce_sum(
              input_tensor=tf.math.log(tlinalg.diag_part(lemma_factor))
          )
      )
  
      logging.log(logging.DEBUG, f"Log-determinant of Q={Q.logdet}")
      logging.log(logging.DEBUG, f"sigma logdet={sigma.logdet}")
      lemma_logdet = 2 * tf.reduce_sum(
          input_tensor=tf.math.log(tlinalg.diag_part(lemma_factor))
      )
      logging.log(
          logging.DEBUG,
          f"lemma factor logdet={lemma_logdet}",
      )
  
      # A^T Sigma^{-1}
      Atrp_Sinv = tf.matmul(A, sigma._prec, transpose_a=True)
      # (Qinv + A^T Sigma^{-1} A)^{-1} A^T Sigma^{-1}
      prod_term = tlinalg.cholesky_solve(lemma_factor, Atrp_Sinv)
  
      solve = tf.matmul(
          sigma.solve(
              scaled_I(1.0, sigma.size) - tf.matmul(A, prod_term)
          ),
          x,
      )
  
      return solve, logdet
  
file_path: brainiak/brainiak/matnormal/matnormal_likelihoods.py
imports: import tensorflow
  from tensorflow import linalg
  from .utils import scaled_I
  import logging
label_desc: Use image node_id 2 for calling a global function with example usage: solve_det_marginal(x, sigma, A, Q) and returns: solve,  logdet