Lines, Circles, Ellipses
Geometry of the Singular Value Decomposition
In Brief: Maybe you remember your professor saying “a symmetric matrix has orthogonal eigenvectors”. Or maybe you remember her saying, “every matrix can be decomposed into a diagonal matrix sandwiched between a pair of unitary matrices”. (Or maybe you don't.)
These two seemingly esoteric ideas are behind a large chunk of modern technology. They come up repeatedly in various forms: as the singular value decomposition (SVD), or as principal component analysis (PCA). As the Karhuenen Loeve or Hoteling transforms. They show up in multivariate gaussian distributions, stress-strain analysis of materials and in quantum orbitals. They are how YouTube knows what you want to watch before you do, and how Tesla autodrives their cars.
But what are these seemingly esoteric ideas? Like, … really? We will teach you in two hours. Down to it's very essence, it's all about geometry. To be more precise, it's about straight lines, and circles. And ellipses.
Target Audience: This lecture is for computer scientists, physicists, engineers, mathematicians and statisticians. And really sharp high school kids.
Course Topics: In the first two hours, we will teach some basic linear algebra, and then the singular value decomposition (SVD), and some applications. We will teach these without taking the traditional eigendecomposition route. In the next two hours (optional - for those with advanced appetite) we will teach a surprising new reformulation of the eigenvectors of symmetric matrices theorem. And some applications.
Prerequisites: High school mathematics. A very basic understanding of linear algebra is useful but not absolutely necessary.
Location: PLEASE NOTE. This lecture will be held in a (yet to be finalized) location in Sunnyvale, California.
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