Subsection 3.1.3 What you will learn¶
This chapter focuses on the QR factorization as a method for computing an orthonormal basis for the column space of a matrix.
Upon completion of this week, you should be able to
Relate Gram-Schmidt orthogonalization of vectors to the QR factorization of a matrix.
Show that Classical Gram-Schmidt and Modified Gram-Schmidt yield the same result (in exact arithmetic).
Compare and contrast the Classical Gram-Schmidt and Modified Gram-Schmidt methods with regard to cost and robustness in the presence of roundoff error.
Derive and explain the Householder transformations (reflections).
Decompose a matrix to its QR factorization via the application of Householder transformations.
Analyze the cost of the Householder QR factorization algorithm.
Explain why Householder QR factorization yields a matrix \(Q \) with high quality orthonormal columns, even in the presence of roundoff error.