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Unit 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.