code: def _matrix_root_eigen(
      A: Tensor,
      root: int,
      epsilon: float = 0.0,
      inverse: bool = True,
      perturb: bool = True,
  ) -> Tuple[Tensor, Tensor, Tensor]:
      """Compute matrix (inverse) root using eigendecomposition of symmetric positive (semi-)definite matrix.
  
              A = Q L Q^T => A^{1/r} = Q L^{1/r} Q^T OR A^{-1/r} = Q L^{-1/r} Q^T
  
      Assumes matrix A is symmetric.
  
      Args:
          A (Tensor): Square matrix of interest.
          root (int): Root of interest. Any natural number.
          epsilon (float): Adds epsilon * I to matrix before taking matrix root. (Default: 0.0)
          inverse (bool): Returns inverse root matrix. (Default: True)
          perturb (bool): Perturbs matrix eigenvalues to ensure it is (practically) positive semi-definite. (Default: True)
  
      Returns:
          X (Tensor): (Inverse) root of matrix. Same dimensions as A.
          L (Tensor): Eigenvalues of A.
          Q (Tensor): Orthogonal matrix consisting of eigenvectors of A.
  
      """
  
      # check if root is positive integer
      if root <= 0:
          raise ValueError(f"Root {root} should be positive!")
  
      # compute matrix power
      alpha = 1 / root
      if inverse:
          alpha = -alpha
  
      # check if matrix is scalar
      if len(A.shape) == 0 or (len(A.shape) == 1 and A.shape[0] == 1):
          return A**alpha, A, torch.tensor(1.0)
  
      # check matrix shape
      if len(A.shape) != 2:
          raise ValueError("Matrix is not 2-dimensional!")
      elif A.shape[0] != A.shape[1]:
          raise ValueError("Matrix is not square!")
  
      # compute eigendecomposition and compute minimum eigenvalue
      L, Q = torch.linalg.eigh(A)
      lambda_min = torch.min(L)
  
      # perturb eigenvalues (if necessary)
      if perturb:
          L += -torch.minimum(lambda_min, torch.tensor(0.0))
  
      # add epsilon
      L += epsilon
  
      # compute inverse preconditioner
      X = Q * L.pow(alpha).unsqueeze(0) @ Q.T
  
      return X, L, Q
  
file_path: bert-ultrafast/training/cramming/backend/optimizers/shampoo/matrix_functions.py
imports: import enum
  import logging
  from typing import Tuple
  import torch
  from torch import Tensor
label_desc: Use image node_id 2 for calling a global function with example usage: _matrix_root_eigen(A, root, epsilon, inverse, perturb) and returns: X,  L,  Q, unknown,  A,  torch