[Normal equations, QR decomposition, Householder reflections, (a) Let A E R™×n with m > n. Show that ATA is positive definite if and only if A has full column rank.1 (b) Let QR = [A b] e Rm×(n+1) be the reduced QR decomposition of the matrix that has a columns the columns of A and additional the column b. Assume the diagonal of Ř is positive. The matrix R has the the elements R r Ř. with RE R"Xn,r E R", p E R. Show that for the least-squares problem min || Ax – b||2, it holds Rx =r and p = ||Ax – b||2.
[Normal equations, QR decomposition, Householder reflections, (a) Let A E R™×n with m > n. Show that ATA is positive definite if and only if A has full column rank.1 (b) Let QR = [A b] e Rm×(n+1) be the reduced QR decomposition of the matrix that has a columns the columns of A and additional the column b. Assume the diagonal of Ř is positive. The matrix R has the the elements R r Ř. with RE R"Xn,r E R", p E R. Show that for the least-squares problem min || Ax – b||2, it holds Rx =r and p = ||Ax – b||2.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![1. [Normal equations, QR decomposition, Householder reflections,
(a) Let A e Rmxn with m > n. Show that ATA is positive definite if and only if A has full
column rank.
(b) Let QR = [A b] € Rmx(n+1) be the reduced QR decomposition of the matrix that has a
columns the columns of A and additional the column b. Assume the diagonal of R is positive.
%3D
The matrix R has the the elements
[R
Ř
with RE R"Xn, r E R", p E R. Show that for the least-squares problem
min || Ax – b||2 ,
it holds Rx =r and p=
|| Ax – b||2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0f75cb18-27c6-41ae-978e-61a4c92df6dc%2Fae7fbbb4-7d79-4497-afb5-db3ff0b1f6da%2Fd9ny1tk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. [Normal equations, QR decomposition, Householder reflections,
(a) Let A e Rmxn with m > n. Show that ATA is positive definite if and only if A has full
column rank.
(b) Let QR = [A b] € Rmx(n+1) be the reduced QR decomposition of the matrix that has a
columns the columns of A and additional the column b. Assume the diagonal of R is positive.
%3D
The matrix R has the the elements
[R
Ř
with RE R"Xn, r E R", p E R. Show that for the least-squares problem
min || Ax – b||2 ,
it holds Rx =r and p=
|| Ax – b||2.
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