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In this section we’re going to look at a way to “decompose”
or “factor” an 
matrix as follows.
Proof : The proof
here will consist of actually constructing Q
and R and showing that they in fact
do multiply to give A.
Okay, let’s start with A
and suppose that it’s columns are given by ,
,
… , . Also suppose that we perform the Gram-Schmidt process on these
vectors and arrive at a set of orthonormal vectors ,
,
… , . Next, define Q (yes, the Q in the theorem
statement) to be the matrix whose columns are ,
,
… , and so Q
will be a matrix with orthonormal columns.
We can then write A and Q as,
Next, because each of the ’s are in we know from Theorem 2 of the previous
section that we can write each as a linear combination of ,
,
… , in the following manner.
Next, define R
(and yes, this will eventually be the R
from the theorem statement) to be the matrix defined as,
Now, let’s examine the product, QR.
From the section on Matrix Arithmetic we know that the jth column of this product is
simply Q times the jth column of R.
However, if you work through a couple of these you’ll see that when we
multiply Q times the jth column of R we arrive at the formula for that we’ve got above. In other words,
So, we can factor A
as a product of Q and R and Q has the correct form. Now
all that we need to do is to show that R
is an invertible upper triangular matrix and we’ll be done. First, from the Gram-Schmidt process we know
that is orthogonal to ,
,
… , . This means that all the inner products below
the main diagonal must be zero since they are all of the form with .
Now, we know from Theorem 2 from the Special
Matrices section that a triangular matrix will be invertible if the main
diagonal entries, ,
are non-zero. This is fairly easy to
show. Here is the general formula for from the Gram-Schmidt process.
Recall that we’re assuming that we found the orthonormal ’s and so each of these will have a
norm of 1 and so the norms are not needed in the formula. Now, solving this for gives,
Let’s look at the diagonal entries of R. We’ll plug in the formula
for into the inner product and do some rewriting
using the properties of the inner product.
However the are orthonormal basis vectors and so we know
that
Using these we see that the diagonal entries are nothing
more than,
So, the diagonal entries of R are non-zero and hence R
must be invertible.
So, now that we’ve gotten the proof out of the way let’s
work an example.
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Example 1 Find
the QR-decomposition for the
matrix,
Solution
The columns
from A are,
We performed
Gram-Schmidt on these vectors in Example 3 of the previous
section. So, the orthonormal vectors
that we’ll use for Q are,
and the matrix Q is,
The matrix R is,
So, the QR-Decomposition for this matrix is,
We’ll leave it to you to verify that this multiplication
does in fact give A.
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