This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: this to make arithmetic easier. EXAMPLE. Find an orthogonal basis for the column space of A = 1 6 6 38 3 12 6 143 . What would happen if, say, the third column of A were a linear combination of the ﬁrst two columns? Suppose A is a p × r matrix with linearly independent columns. GramSchmidt the columns of A , in order, and then normalize the columns, that is, divide each column by its length, to produce the columns of a matrix Q . What can we say about Q T Q ? Can we say that Q is invertible? Explain how we know there must be a matrix R such that QR = A . What can we say about R ? R = Q T A is an invertible upper triangular matrix. What we have here is called the QR factorization of A . It is heavily used in numerical methods for ﬁnding eigenvalues (p. 318) and solving equations (p. 414). HOMEWORK: SECTION 6.4...
View
Full
Document
This note was uploaded on 03/06/2010 for the course M 340L taught by Professor Pavlovic during the Spring '08 term at University of Texas.
 Spring '08
 PAVLOVIC

Click to edit the document details