Chapter 7: Symmetric Matrices and Quadratic Forms

Class Notes

• Lay, Section 7.1, Diagonalization of Symmetric Matrices (pdf)
• Lay, Section 7.2, Quadratic Forms (pdf)
• Lay, Section 7.3, Constrained Optimization (pdf)
• Lay, Section 7.4, The Singular Value Decomposition (pdf)
• Lay, Section 7.5, Applications to Image Processing and Statistics (pdf)

Homework Exercises

• Lay, Section 7.1, pp 454--455 -- 1, 11, 17, 21, 23, 29, 33 (spectral decomposition), 35(orthogonal projection), 37, 39 (orthogonal diagonalization)
• Lay, Section 7.2, pp 462--463 -- 1, 3, 5, 7, 13, 15 (diagonalize a quadratic form), 23, 24 (classification of 2x2 quadratic forms), 25 (B^TB), 27 (eigenvalues of A+B)
• Lay, Section 7.3, pp 470--471 -- 1, 5, 9, 11, 13, 15, 17 (constrained optimization)
• Lay, Section 7.4, pp 481--482 -- 3, 7, 9, 13, 15 (fundamental subspaces), 17, 18, 19, 20, 27 (singular value decomposition), 29 (condition number)
• Lay, Section 7.5, pp 489--490 -- 1 (covariance matrix), 3 (principal components), 5 (Landsat image), 7 (variance), 9 (testing college students)

Case Study and Applications

Download the supporting pdf files and Mathematica notebooks from the Lay Linear Algebra web site.

• The Singular Value Decomposition and Image Processing
• Conic Sections and Quadratic Surfaces
• Extrema for Functions of Several Variables