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- A = QR
There are several methods for actually computing the QR decomposition, such as by means of Givens rotations, Householder transformations, or the Gram-Schmidt decomposition. Each has a number of advantages and disadvantages. (The matrices Q and R are not uniquely determined, so different methods may produce different decompositions.)
Computing QR by means of Gram-Schmidt
Recall the Gram-Schmidt method, with the vectors to be considered in the process as columns of the matrix . Then
Naturally then, we rearrange the equations so the ais are the subject, to get the following
Each of these projections of the vectors onto one of these ej are merely the inner product of the two, since the vectors are normed.
Now these equations can be written in matrix form, viz.,
But the product of each row and column of the matrices above give us a respective column of A that we started with, and together, they give us the matrix A, so we have factorized A into an orthogonal matrix Q (the matrix of eks), via Gram Schmidt, and the obvious upper triangular matrix as a remainder R.
Alternatively, can be calculated as follows:
Recall that Then, we have
Note that and QQT = I, so QT = Q - 1.
Consider the decomposition of
Recall the orthogonal matrix Q such that
Then, we can calculate Q by means of Gram-Schmidt as follows:
Thus, we have
Considering numerical errors of finite precision operation in MATLAB, we have that
Computing QR by means of Householder reflections
A Householder reflection (or Householder transformation) is a transformation that takes a vector and reflects it about some plane. We can use this property to calculate the QR factorization of a matrix.
Q can be used to reflect a vector in such a way that all coordinates but one disappear. Let x be an arbitrary m-dimensional column vector of length |α| (for numerical reasons α should get the same sign as the first coordinate of x).
Then, where e1 is the vector (1,0,...,0)T, and || || the euclidean norm, set
Q is a Householder matrix and
This can be used to gradually transform an m-by-n matrix A to upper triangular form. First, we multiply A with the Householder matrix Q1 we obtain when we choose the first matrix column for x. This results in a matrix QA with zeros in the left column (except for the first line).
This can be repeated for A' resulting in a Housholder matrix Q'2. Note that Q'2 is smaller than Q1. Since we want it really to operate on Q1A instead of A' we need to expand it to the upper left, filling in a 1, or in general:
After t iterations of this process, t = min(m - 1,n),
is a upper triangular matrix. So, with
A = QR is a QR decomposition of A.
This method has greater numerical stability than using the Gram-Schmidt method above. .
Let us calculate the decomposition of
We need to find a reflection that takes the vector a1 = (12,6, - 4)T to .
Now, u = ( - 2,6, - 4)T and , and then
So we already have almost a triangular matrix. We only need to zero the (3, 2) entry.
Take the (1, 1) minor, and then apply the process again to
By the same method as above, we obtain the matrix of the Householder transformation to be,
after performing a direct sum with 1 to make sure the next step in the process works properly.
Now, we find
The matrix Q is orthogonal and R is upper triangular, so A = QR is the required QR-decomposition.
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