QR Factorization of A: Simple Procedure

In summary, the QR factorization of A is obtained by using Schmidt orthogonalization to find Q and recording the coefficients to get R. Q must be orthogonal and R must be upper triangular. There may be a simpler procedure, but this method involves assigning Q and P matrices and using Mathematica to find P. However, there may be some confusion or misunderstanding about the process.
  • #1
rbwang1225
118
0

Homework Statement


Find the QR factorization of A = {1, 1}, {-1, 1}

The Attempt at a Solution


I just don't know the procedure.
I know it means that I need find Q and R such that A=QR, Q be orthogonal, and R be upper triangular.
It may be solved by assign Q = {a, b},{c, d}, where ##Q^TQ=1##
and P = {e, f}, {0, g}
But, as I run Mathematica, it gives me P that the left-down side entry is nonzero.
Is there any point that I misunderstood?
And is there any way to simplify the procedure?
P.S. Please forgive me that I don't know how to "type" a matrix.
Regards.
 
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  • #2
Q is obtained by Schmidt othnormalization, record the coefficients to get R.
 

1. What is QR factorization of A?

QR factorization of A is a mathematical procedure that decomposes a matrix A into an orthogonal matrix (Q) and an upper triangular matrix (R). It is a useful tool in solving systems of linear equations and in data analysis.

2. Why is QR factorization useful?

QR factorization simplifies calculations involving matrices and makes solving systems of linear equations more efficient. It is also used in data analysis for dimension reduction and identifying important features or patterns in the data.

3. How is QR factorization performed?

QR factorization is typically performed using the Gram-Schmidt process, which involves orthogonalizing the columns of A to create the matrix Q, and then using Q to find the upper triangular matrix R. Other methods, such as Householder and Givens transformations, can also be used to perform QR factorization.

4. What are the applications of QR factorization?

QR factorization has various applications in mathematics, engineering, and data analysis. It is used in solving systems of linear equations, least squares approximation, eigenvalue computations, and in algorithms for data compression and dimension reduction.

5. Can QR factorization fail?

Yes, QR factorization can fail in certain cases. For example, if the matrix A is singular or ill-conditioned, the decomposition may not be possible. Additionally, rounding errors in the calculations can also affect the accuracy of the decomposition.

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