Unit 9 of 10
Numerical Linear Algebra
- 5 lessons
- Videos 46 to 50
This unit separates the sensitivity of a problem from the errors a method adds, and uses that distinction to compare ways of solving linear systems.
Matrix norms and condition numbers measure how far a small change in the data can move a solution, which explains why a small residual need not mean an accurate answer. Householder QR, the normal equations and Cholesky factorization are compared for least-squares and positive definite systems, and power iteration shows when repeated multiplication finds a dominant eigenvector and when it fails.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Size, Conditioning and Error
Matrix Norms and Condition Numbers
How induced and Frobenius norms measure the size of a matrix, why the spectral norm is the largest singular value, and how the condition number bounds the change in a solution when the right-hand side changes.
Video 46
Residuals, Forward Error and Backward Error
Why a small residual need not mean an accurate solution, how forward and backward error answer different questions, how the condition number links them, and what backward stability and numerical rank mean.
Video 47
Reliable Factorization Choices
Householder QR Factorization
How length-preserving Householder reflectors clear a column, why the stable sign choice avoids cancellation in each reflector, and how full and reduced QR give the least-squares solution by a triangular solve and its residual.
Video 48
QR, Normal Equations and Cholesky
Why forming the Gram matrix squares the condition number of a full-column-rank matrix, and how QR, normal equations, Cholesky, partial pivoting and the SVD fit different structures and ranks.
Video 49
Power Iteration
Power Iteration
How power iteration finds a dominant eigenvector of a diagonalizable matrix, what sets its convergence rate, how the start or equal-modulus eigenvalues make it fail, and when residuals bound eigenvalue errors.
Video 50