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HomeMIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018Lecture 7: Eckart-Young: The Closest Rank k Matrix to A
MIT 18.065 Matrix Methods in Data Analysis, Signal Processing, and Machine Learning, Spring 2018
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Lecture 7: Eckart-Young: The Closest Rank k Matrix to A
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Lecture 8: Norms of Vectors and Matrices
Description
In this lecture, Professor Strang reviews Principal Component Analysis (PCA), which is a major tool in understanding a matrix of data. In particular, he focuses on the Eckart-Young low rank approximation theorem.
Summary
\(A_k = \sigma_1 u_1 v^{\mathtt{T}}_1 + \cdots + \sigma_k u_k v^{\mathtt{T}}_k\) (larger \(\sigma\)’s from \(A = U\Sigma V^{\mathtt{T}}\))
The norm of \(A - A_k\) is below the norm of all other \(A - B_k\).
Frobenius norm squared = sum of squares of all entries
The idea of Principal Component Analysis (PCA)
Related section in textbook: I.9
Instructor: Prof. Gilbert Strang
