**Course Title:** MATLAB Programming: Applications in Engineering, Data Science, and Simulation **Section Title:** Numerical Computation and Linear Algebra **Topic:** Eigenvalues, eigenvectors, and singular value decomposition (SVD) ### Overview In this topic, we will delve into three fundamental concepts in linear algebra: eigenvalues, eigenvectors, and singular value decomposition (SVD). These concepts have numerous applications in various fields, including engineering, data science, and machine learning. You will learn how to find eigenvalues and eigenvectors of a matrix, and how to compute the SVD of a matrix using MATLAB. ### Eigenvalues and Eigenvectors An eigenvalue is a scalar that represents how much a matrix changes a vector when multiplied by that vector. An eigenvector is a non-zero vector that, when multiplied by a matrix, results in a scaled version of itself. The equation for finding eigenvalues and eigenvectors is: AX = λX where A is the matrix, X is the eigenvector, and λ (lambda) is the eigenvalue. To find the eigenvalues and eigenvectors of a matrix A in MATLAB, you can use the `eig()` function: ```matlab A = [2 1; 1 1]; % Create a 2x2 matrix [V, D] = eig(A); % Compute eigenvalues and eigenvectors % V is a matrix where each column is an eigenvector % D is a diagonal matrix where each element is an eigenvalue ``` ### Singular Value Decomposition (SVD) Singular value decomposition (SVD) is a powerful technique for matrix factorization. SVD decomposes a matrix A into three matrices: U, Σ, and V, such that A = UΣV', where: * U is an orthogonal matrix (U'U = I) * Σ is a diagonal matrix containing the singular values of A * V is an orthogonal matrix (V'V = I) The SVD of a matrix A can be computed in MATLAB using the `svd()` function: ```matlab A = [2 1; 1 1; 1 2]; % Create a 3x2 matrix [U, S, V] = svd(A); % Compute SVD ``` ### Applications of Eigenvalues, Eigenvectors, and SVD These concepts have numerous applications in various fields, including: * **Stability analysis**: Eigenvalues are used to determine the stability of a system. * **Data compression**: SVD is used for image and signal compression. * **Machine learning**: Eigenvalues and eigenvectors are used in principal component analysis (PCA) and other machine learning algorithms. * **Cryptography**: SVD is used to break certain encryption algorithms. ### Example Problem Compute the eigenvalues and eigenvectors of the matrix: A = [2 2; 2 2] ```matlab A = [2 2; 2 2]; [V, D] = eig(A); eigenvalues = diag(D); fprintf('Eigenvalues: %.2f, %.2f\n', eigenvalues(1), eigenvalues(2)); ``` Compute the SVD of the matrix: A = [2 1; 1 1; 1 2] ```matlab A = [2 1; 1 1; 1 2]; [U, S, V] = svd(A); singular_values = diag(S); fprintf('Singular values: %.2f, %.2f\n', singular_values(1), singular_values(2)); ``` ### Conclusion In this topic, we covered the fundamental concepts of eigenvalues, eigenvectors, and singular value decomposition (SVD). We learned how to compute these using MATLAB and explored some of their applications. By mastering these concepts, you will be able to tackle more advanced problems in linear algebra and its applications. **Practical Takeaways** * Compute eigenvalues and eigenvectors using `eig()` * Compute SVD using `svd()` * Apply eigenvalues, eigenvectors, and SVD to solve problems in stability analysis, data compression, machine learning, and cryptography **External Resources** * [Wikipedia: Eigenvalues and eigenvectors](https://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors) * [Wikipedia: Singular value decomposition](https://en.wikipedia.org/wiki/Singular_value_decomposition) * [MathWorks: `eig()` documentation](https://www.mathworks.com/help/matlab/ref/eig.html) * [MathWorks: `svd()` documentation](https://www.mathworks.com/help/matlab/ref/svd.html) **Do you have any questions or need help with this material? Please leave a comment or ask for help below.** In the next topic, we will cover numerical integration and differentiation.