1. Generate a random sparse matrix with a specified density and size.
2. Create a Hilbert matrix of size n and demonstrate its properties.
3. Construct a matrix representing a random geometric graph with n nodes.
4. Generate a circulant matrix from a given list of values.
5. Create a Vandermonde matrix from a list of numbers and demonstrate its properties.
6. Construct a block diagonal matrix from multiple matrices.
7. Generate a Toeplitz matrix from a specific first row and first column.
8. Create a companion matrix from a polynomial representation.
9. Construct a matrix representation of a Markov chain from transition probabilities.
10. Generate a symmetric positive definite matrix.
11. Implement matrix exponentiation to raise a matrix to a power.
12. Perform singular value decomposition (SVD) on a given matrix.
13. Implement QR decomposition and explain its applications.
14. Create a function to calculate the eigenvalues and eigenvectors of a matrix.
15. Implement the LU decomposition algorithm for matrix factorization.
16. Compute the Cholesky decomposition of a positive definite matrix.
17. Calculate the condition number of a matrix and explain its significance.
18. Implement the Jacobi method for solving linear systems.
19. Solve a system of linear equations using the Gauss-Seidel method.
20. Perform a least squares approximation using matrix operations.
21. Implement the Hungarian algorithm for solving assignment problems.
22. Use a matrix representation to find the shortest path in a weighted graph.
23. Implement Dijkstra’s algorithm on a matrix representation of a graph.
24. Use A* search to find the shortest path in a matrix grid.
25. Create a function to detect cycles in a directed matrix (graph).
26. Implement a function to find strongly connected components in a directed graph using a matrix.
27. Count the number of unique paths in a matrix from top-left to bottom-right.
28. Find the longest increasing path in a matrix of integers.
29. Create a function to search for a word in a 2D character grid.
30. Determine if a matrix contains any path from one corner to another without crossing barriers.
31. Implement an affine transformation on a matrix representing image data.
32. Create a function to perform perspective projection on 3D points represented in a matrix.
33. Rotate a matrix around a specified point and return the modified matrix.
34. Implement bilinear interpolation on an image matrix.
35. Create a function to perform geometric transformations (scaling, rotating, translating).
36. Implement image blurring using a convolution matrix.
37. Perform edge detection on an image matrix using Sobel filters.
38. Create a Gaussian blur effect on an image matrix.
39. Implement a function to apply a sharpening filter to an image matrix.
40. Create a function to compute the Fourier transform of a matrix.