Unit III: Positive Definite Matrices and Applications

Singular Value Decomposition

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Session Overview

Figure excerpted from 'Introduction to Linear Algebra' by G.S. Strang

If A is symmetric and positive definite, there is an orthogonal matrix Q for which A = Q_Λ_QT. Here Λ is the matrix of eigenvalues. Singular Value Decomposition lets us write any matrix A as a product U_Σ_VT where U and V are orthogonal and Σ is a diagonal matrix whose non-zero entries are square roots of the eigenvalues of ATA. The columns of U and V give bases for the four fundamental subspaces.

Session Activities

Lecture Video and Summary

Suggested Reading

  • Read Section 6.7 in the 4th edition or Section 7.1 and 7.2 in the 5th edition.

Problem Solving Video

Check Yourself

Problems and Solutions

Work the problems on your own and check your answers when you’re done.

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Course Info

Learning Resource Types

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grading Exams with Solutions
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theaters Recitation Videos
assignment_turned_in Problem Sets with Solutions
laptop_windows Simulations
groups Course Introduction
co_present Instructor Insights