code: def evaluate_coefficients(
      pdf: pd.DataFrame, alpha_values: Iterable[Float], n_cov: int
  ) -> NDArray[Float]:
      """
      Solves the system (XTX + Ia)^-1 * XtY for each of the a values in alphas.  Returns the resulting coefficients.
  
      Args:
           pdf : Pandas DataFrame for the group
           alpha_values : Array of alpha values (regularization strengths)
           n_cov: Number of covariate columns on the left-most side of matrix X.  These are regularized with a constant
           value of alpha = 1, regardless of the alpha value being used for the rest of the matrix X.
  
      Returns:
          Matrix of coefficients of size [number of columns in X] x [number of labels * number of alpha values]
      """
      XtX = np.stack(pdf["xtx"].array)
      XtY = np.stack(pdf["xty"].array)
      diags = [
          np.concatenate(
              [np.ones(n_cov), np.ones(XtX.shape[1] - n_cov) * a]
          )
          for a in alpha_values
      ]
      return np.column_stack(
          [(np.linalg.inv(XtX + np.diag(d)) @ XtY) for d in diags]
      )
  
imports: import itertools
  import math
  import re
  import warnings
  from typing import Any, Dict, Iterable, List, Tuple
  import numpy
  import pandas
  import pyspark.sql.functions
  import scipy.optimize
  from nptyping import Float, Int, NDArray
  from pyspark.sql import DataFrame, Row
  from pyspark.sql.window import Window
  from typeguard import typechecked
label_desc: Use image node_id 8 for calling a global function with example usage: evaluate_coefficients(pdf, alpha_values, n_cov) and returns: np