Exercises¶

Use the button below to open a template in Compiler Explorer with the Eigen library included. You can use this template to complete the exercises below.

• Open solution template in Compiler explorer
• Solutions
• Use the template folder for local development

How to use this guide¶

In a 6-hour course, we recommend:

• Lab sessions 1–2 (10:20–11:00, 13:00–13:40): Exercises 1–7 (Basic & Intermediate)
• Lab session 3 (14:20–15:00): Exercises 8–12 (Advanced operations) or continue with 1–7 if needed
• Self-study / extended practice: Exercises 13–15 (Linear algebra & applications)

Prerequisites: Complete exercises 1–3 before moving to exercises 4–7. Complete 1–7 before attempting 8–15.

Difficulty legend: 🟢 Beginner | 🟡 Intermediate | 🔴 Advanced

Basic Matrix and Vector Operations¶

1. Vector Creation and Access¶

🟢 Beginner | ⏱️ 5 min

• Create a 5-element vector using Eigen and initialize it with values 1 through 5.
• Print the vector's elements and demonstrate how to access individual elements.

2. Matrix Initialization¶

🟢 Beginner | ⏱️ 5 min

• Create a 3x3 matrix with all elements set to zero.
• Initialize a 2x4 matrix with specific values of your choice.
• Print both matrices to verify their contents.

3. Basic Math Functions¶

🟢 Beginner | ⏱️ 10 min

• Create a vector with values ranging from -5 to 5.
• Apply the absolute value, square root (for positive elements), and exponential functions.
• Calculate the sum, mean, minimum, and maximum values of the vector.

4. Coefficient-wise Operations¶

🟡 Intermediate | ⏱️ 10 min

• Create two vectors of the same length.
• Perform element-wise multiplication, division, and power operations.
• Calculate the dot product and compare with element-wise multiplication.

5. Matrix Operations¶

🟡 Intermediate | ⏱️ 10 min

• Create two 3x3 matrices with different values.
• Perform and print the results of addition, subtraction, and element-wise multiplication.
• Calculate and print the matrix product of the two matrices.

6. Vector-Matrix Operations¶

🟡 Intermediate | ⏱️ 10 min

• Create a 3x3 matrix and a 3-element vector.
• Multiply the matrix by the vector and explain the result.
• Experiment with different ways to perform the multiplication.

7. Special Matrix Operations¶

🟡 Intermediate | ⏱️ 15 min

• Create a 4x4 matrix and calculate its determinant, inverse, and transpose.
• Verify that A * A^-1 equals the identity matrix (within numerical precision).
• Explore how to check if a matrix is singular and handle potential errors.

Advanced Matrix Operations¶

8. Block Operations¶

🟡 Intermediate | ⏱️ 15 min

• Create a 4x4 matrix with sequential numbers (1-16).
• Extract the 2x2 block in the top-left corner.
• Extract the 2x2 block in the bottom-right corner.
• Replace the bottom-left 2x2 block with a new 2x2 matrix.

9. Row and Column Operations¶

🟡 Intermediate | ⏱️ 15 min

• Create a 3x4 matrix with random values.
• Extract the second row and third column.
• Replace the first row with new values.
• Swap two columns of your choice.

10. Resizing and Reshaping¶

🟡 Intermediate | ⏱️ 10 min

• Create a 2x3 matrix and then resize it to a 3x2 matrix.
• Create a 6-element vector and reshape it into a 2x3 matrix.
• Discuss what happens to the elements during these operations.

11. Concatenation¶

🟡 Intermediate | ⏱️ 15 min

• Create two 2x2 matrices.
• Horizontally concatenate them to form a 2x4 matrix.
• Vertically concatenate them to form a 4x2 matrix.
• Create and demonstrate a function that can concatenate matrices of compatible dimensions.

12. Advanced Slicing¶

🔴 Advanced | ⏱️ 20 min

• Create a 5x5 matrix with sequential values.
• Extract non-contiguous rows and columns (for example rows 1, 3, 4 and columns 0, 2).
• Extract a diagonal or anti-diagonal of the matrix.
• Create a tiled pattern by repeating a smaller matrix.

13. Linear Algebra Operations¶

🔴 Advanced | ⏱️ 20 min

• Create a system of linear equations represented as Ax = b.
• Solve the system using Eigen's solver capabilities.
• Verify your solution by substituting it back into the original equations.

14. Eigenvalues and Eigenvectors¶

🔴 Advanced | ⏱️ 20 min

• Create a symmetric 3x3 matrix.
• Calculate its eigenvalues and eigenvectors.
• Verify that A v = lambda v for each eigenvalue-eigenvector pair.

Practical Applications¶

15. Image Processing Basics¶

🔴 Advanced | ⏱️ 30 min

• Represent a grayscale image as an Eigen matrix.
• Implement basic operations like brightness adjustment and contrast enhancement.
• Apply simple filters (for example blur) using matrix operations.

Img::ImageGd image;

std::cout << "Loading image..." << std::endl;

if (!Img::load("images/half-moon-986269.png", image))
{
    std::cout << "Error loading image" << std::endl;
    return 1;
}

MatrixXd matrix = image.as_matrix();

Img::ImageGd copyToImage(const Eigen::MatrixXd &mat)
{
    ImageGd image;
    image.resize(mat.rows(), mat.cols());
    for (int i = 0; i < mat.rows(); i++)
        for (int j = 0; j < mat.cols(); j++)
            image(i, j) = mat(i, j);

    return image;
}

if (!Img::save("output.png", copyToImage(matrix)))
{
    std::cout << "Error saving image" << std::endl;
    return 1;
}

Eigen::MatrixXd createGaussianKernel(int size, double sigma)
{
    if (size % 2 == 0)
    {
        size++; // Make sure kernel size is odd
    }

    Eigen::MatrixXd kernel(size, size);
    int center = size / 2;
    double sum = 0.0;

    // Fill the kernel with Gaussian values
    for (int y = 0; y < size; y++)
    {
        for (int x = 0; x < size; x++)
        {
            int dx = x - center;
            int dy = y - center;
            float exponent = -(dx * dx + dy * dy) / (2 * sigma * sigma);
            kernel(y, x) = std::exp(exponent) / (2 * std::numbers::pi * sigma * sigma);
            sum += kernel(y, x);
        }
    }

    // Normalize the kernel so its sum equals 1
    kernel /= sum;

    return kernel;
}