QR Fatoration (Linear Algebra) (FIXED)

[leoh]

Homework Statement

Suppose that A is an matrix with linearly independent columns then A can be factored as,
where Q is an matrix with orthonormal columns and R is an invertible upper triangular matrix.

Show why it is Upper triangular matrix, and why the elements of the diagonal are Positive.

Homework Equations

A=QR
A= [x1 x2 ... xk]
R= [r1 r2 ... rk]

The Attempt at a Solution

(Sorry for my bad English, I am from Brazil)x1= (x1.u1)u1+(x1.u2)u2+...+(x1.uk)uk
x2= (x2.u1)u1+(x2.u2)u2+...+(x2.uk)uk
.
.
.
xn= (xn.u1)u1+(xn.u2)u2+...+(xn.uk)uk

now there is an argument, given Xi and Uj, if i<j , Uj and Xi are orthogonals. Which I do not know how to prove it.
So, if this is valid:

x1= (x1.u1)u1
x2= (x2.u1)u1+(x2.u2)u2
.
.
.
xk= (xk.u1)u1+(xk.u2)u2+...+(xk.uk)ukwhich is triangular. Now I do not now why the product (x1.u1), (xk.uk) is always positive.

 

[leoh]

nobody?