Q1. Scenario: In Principal Component Analysis (PCA), you want to reduce 1000-dimensional data to 2 dimensions while preserving variance. What concept from linear algebra helps you find the most important directions?
Eigenvectors of the covariance matrix represent the principal directions (principal components). The eigenvalues indicate how much variance is explained by each eigenvector. Sorting eigenvectors by descending eigenvalues gives the most important components. The top 2 eigenvectors project the data to 2D while maximizing variance.
Q2. Scenario: A search engine uses PageRank to rank web pages. The PageRank vector is an eigenvector of the link matrix. Explain how eigenvalues determine convergence and why the largest eigenvalue is 1.
PageRank is the eigenvector corresponding to eigenvalue 1 of a stochastic matrix. The largest eigenvalue of a stochastic matrix is 1, ensuring that the iterative power method converges to a unique stationary distribution. Other eigenvalues with magnitude <1 determine the rate of convergence (the second eigenvalue defines the mixing time).
Q3. Scenario: In facial recognition systems (Eigenfaces), how are eigenvectors used to represent faces?
Each face image is treated as a vector. The eigenvectors of the covariance matrix of face images are called eigenfaces". They form a basis of the face space. Any face can be approximated by a linear combination of a few eigenfaces reducing dimensionality while capturing essential features for recognition.
Q4. Scenario: In dynamic systems a system of differential equations dx/dt = A·x. How do eigenvalues and eigenvectors help determine stability?
The general solution is x(t) = c1·v1·e^(λ1·t) + c2·v2·e^(λ2·t) + ... where λ are eigenvalues v eigenvectors. If any λ has positive real part the system is unstable (grows exponentially). If all λ have negative real parts it is stable (decays to zero). This is crucial in control theory population dynamics and circuit analysis.
Q5. Scenario: In quantum mechanics observable quantities (like energy) are represented by matrices and possible measurement outcomes are eigenvalues. How would you find the possible energy levels of a system?
"Solve the eigenvalue equation H·ψ = E·ψ where H is the Hamiltonian matrix (energy operator). The eigenvalues E are the possible energy levels and eigenvectors ψ are the corresponding quantum states. This is the foundation of quantum computing and atomic physics.