Notes

# Do not import any additional libraries for this section
# Do not change any code outside of this area marked with =============================

# ====================== YOUR CODE HERE ======================
# DO NOT use any other import statements for this question

# Step 1: Calculate the mean of each column
mean = np.mean(X, axis=0)

# Step 2: Centralize the data by subtracting the mean
X_C = X - mean

# Step 3: Calculate the covariance matrix
C = np.cov(X_C, rowvar=False)

# Step 4: Calculate the eigenvalues and eigenvectors
eigenvalues, eigenvectors = np.linalg.eigh(C)

# Step 5: Select the top 10 principal components (eigenvectors corresponding to the top 10 eigenvalues)
# Sort eigenvalues and corresponding eigenvectors
sorted_indices = np.argsort(eigenvalues)[::-1]
top_indices = sorted_indices[:no_of_components]
P = eigenvectors[:, top_indices]

# Step 6: Transform the data using the principal components
X_transformed = np.dot(X_C, P)

# ====================== YOUR CODE HERE ======================

# Reconstruct the original data from the transformed data
X_back = np.dot(X_transformed, P.T) + mean

# Calculate the covariance matrix of the reconstructed data
covariance_matrix_X_back = np.cov(X_back, rowvar=False)

# ============================================================
print("Sum of covariance_matrix_X_back:")
print(np.round(np.sum(covariance_matrix_X_back),2))