Lecture 9 Some Linear Algebra Nicholas Ruozzi University
Lecture 9: Some Linear Algebra Nicholas Ruozzi University of Texas at Dallas
Length of Vectors • 2
Linear Combinations • 3
Linear Independence • Linearly independent vectors 4
Linear Independence • Linearly dependent vectors 5
Linear Independence • Orthogonal vectors 6
Matrix Rank • 7
Orthonormal Vectors • 8
Orthonormal Vectors • 9
Eigenvalues • 10
Eigenvalues of Symmetric Matrices • 11
Examples • 12
Eigen-decomposition • 13
Eigen-decomposition • 14
Eigen-decomposition • 15
Useful Properties of Positive Semidefinite Matrices • 16
Useful Properties of Positive Semidefinite Matrices • 17
Useful Properties of Positive Semidefinite Matrices • 18
Least Square Solutions to Linear Systems • 19
Least Square Solutions to Linear Systems • 20
Least Square Solutions to Linear Systems • 21
Least Square Solutions to Linear Systems • 22
Least Square Solutions to Linear Systems • 23
Least Square Solutions to Linear Systems • 24
Least Square Solutions to Linear Systems • 25
The Positive Semidefinite Matrices • 26
Projection on PSD Matrices • 27
Projection on PSD Matrices • 28
Projection on PSD Matrices • 29
Projection on PSD Matrices • 30
Inner Products for Matrices • 31
Inner Products for Matrices • 32
Projection on PSD Matrices • 33
Projection on PSD Matrices • 34
Projection on PSD Matrices • 35
Semidefinite Programming (SDP) • 36
Semidefinite Programming • 37
Rayleigh-Ritz Theorem • 38
Rayleigh-Ritz Theorem • 39
Computing Eigenvalues • There a variety of methods for computing the eigenvalues/eigenvectors of a matrix • Power iteration: an iterative method that only requires performing matrix multiplication and dot products • QR factorization: based on Gram-Schmidt orthogonalization (we won’t cover it here, take CS 4334: Numerical Analysis!) 40
Power Iteration • 41
- Slides: 41