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@ -383,3 +383,11 @@ Fourier series vs. DFT: If we view the DFT as a [Riemann sum](https://en.wikiped
* Kronecker products, [Hadamard matrices](https://en.wikipedia.org/wiki/Hadamard_matrix), and the mixed-product property (A⊗B)(C⊗D)=AC⊗BD: Kronecker products of unitary matrices are unitary.
* Kronecker products and the [fast Walsh–Hadamard transform (FWHT)](https://en.wikipedia.org/wiki/Fast_Walsh%E2%80%93Hadamard_transform): fast algorithms (FWHT, FFT, …) as *sparse factorizations* of dense matrices via Kronecker products, and equivalently as mono-to-multi-dimensional mappings.
* [Sylvester equations](https://en.wikipedia.org/wiki/Sylvester_equation) and [Lyapunov equations](https://en.wikipedia.org/wiki/Lyapunov_equation): can be solved as ordinary matrix–vector equations via Kronecker products, but this naively requires Θ(n⁶) operations, whereas exploiting the structure gives Θ(n³). Kronecker products are often a nice way to *think* about things but *not* to explicitly compute with.
## Lecture 37 (May 10)
* [Graphs](https://en.wikipedia.org/wiki/Graph_(discrete_mathematics)) and their application to representing how entities are connected — lots of applications, from data mining of social networks or the web, to bioinformatics, to analyzing sparse-matrix algorithms
* Representing graphs via matrices: [incidence matrices E](https://en.wikipedia.org/wiki/Incidence_matrix), [adjacency matrices A](https://en.wikipedia.org/wiki/Adjacency_matrix), and [graph Laplacians L=EᵀE=D-A](https://en.wikipedia.org/wiki/Laplacian_matrix)
* [Graph partitioning](https://en.wikipedia.org/wiki/Graph_partition): spectral partitioning via the [Fiedler vector](https://en.wikipedia.org/wiki/Algebraic_connectivity), corresponding to the **second-smallest** eigenvalue of L
**Further reading:** Textbook sections VI.6–VI.7 and [OCW lecture 35](https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/resources/lecture-35-finding-clusters-in-graphs-second-project-handwriting/). Using incidence matrices to identify cycles and spanning trees in graphs is also covered in 18.06 and in Strang's *Introduction to Linear Algebra* book (5th ed. section 10.1 or 6th ed. section 3.5), as well as in [this interactive Julia notebook](https://github.com/mitmath/1806/blob/1a9ff5c359b79f28c534bf1e1daeadfdc7aee054/notes/Graphs-Networks.ipynb). A popular software package for graph and mesh partitioning is [METIS](https://github.com/KarypisLab/METIS). The Google [PageRank algorithm](https://en.wikipedia.org/wiki/PageRank) is another nice application of linear algebra to graphs, in this case to rank web pages by "importance". Direct methods (e.g. Gaussian elimination) for sparse-matrix problems turn out to be all about analyzing sparsity patterns via graph theory; see e.g. the [book by Timothy Davis](https://epubs.siam.org/doi/book/10.1137/1.9780898718881).