QR Factorization: Show A=LQ, L Triangular & Q Orthogonal

In summary: Since S is orthogonal, its transpose must also be orthogonal. Therefore, S^T=Q, and we can write A=LQ where L is a lower triangular matrix and Q is orthogonal.In summary, for an invertible n x n matrix A, it is possible to write A as A=LQ, where L is a lower triangular matrix and Q is orthogonal. This can be shown by considering the QR factorization of A^T, where Q is orthogonal and L is a lower triangular matrix. The transpose of S, ST, is also orthogonal, and since S^T=Q, we can write A=LQ.
  • #1
Clandry
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Homework Statement


Consider an invertible n x n matrix A. Can you write A as A=LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of #A^T#.

Homework Equations


For QR factorization, Q is orthogonal and R is upper triangular.

The Attempt at a Solution


If we consider the hint, then we can write:
##A^T=S*U## where S is orthogonal matrix and U is some upper triangular matrix.
##(A^T)^T=U^T*S^T##; transpose of upper triangular matrix U is some lower triangular matrix L
##A=L*S^T##

Here is where I get lost. I don't know how to show that S^T=Q. Could someone please give me a hint?
 
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  • #2
Clandry said:

Homework Statement


Consider an invertible n x n matrix A. Can you write A as A=LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of #A^T#.

Homework Equations


For QR factorization, Q is orthogonal and R is upper triangular.

The Attempt at a Solution


If we consider the hint, then we can write:
#A^T+=S*U# where S is orthogonal matrix and U is some upper triangular matrix.
#(A^T)^T=U^T*S^T#; transpose of upper triangular matrix U is some lower triangular matrix L
#A=L*S^T#

Here is where I get lost. I don't know how to show that S^T=Q. Could someone please give me a hint?
Your # symbols are cluttering up your work, making it harder to read than it should be. If you trying to use LaTeX, use two # at the beginning and two more at the end.

Regarding your question, S is orthogonal, right. What about its transpose, ST? Isn't that orthogonal as well?
 
  • #3
Mark44 said:
Your # symbols are cluttering up your work, making it harder to read than it should be. If you trying to use LaTeX, use two # at the beginning and two more at the end.

Regarding your question, S is orthogonal, right. What about its transpose, ST? Isn't that orthogonal as well?
Ah yes! I think I overthought this problem.
 

1. What is QR factorization?

QR factorization is a mathematical technique used to decompose a matrix into two matrices, L and Q, where L is a lower triangular matrix and Q is an orthogonal matrix.

2. How is QR factorization performed?

QR factorization is typically performed using the Gram-Schmidt process or the Householder transformation method. These methods involve a series of mathematical operations that result in the decomposition of the original matrix into L and Q.

3. What is the importance of QR factorization?

QR factorization is important in many applications of linear algebra, such as solving systems of linear equations, finding eigenvalues and eigenvectors, and performing least squares regression. It also allows for more efficient and accurate computations compared to using the original matrix.

4. How can you show that A = LQ in QR factorization?

To show that A = LQ in QR factorization, you can perform the matrix multiplication of L and Q and show that the result is equal to A. This is because the product of a lower triangular matrix and an orthogonal matrix will always result in the original matrix A.

5. Why is L a lower triangular matrix and Q an orthogonal matrix in QR factorization?

In QR factorization, L is a lower triangular matrix because the Gram-Schmidt process or Householder transformation method involves subtracting vectors from the original matrix to create the lower triangular matrix. Q is an orthogonal matrix because it is composed of orthogonal vectors, meaning they are perpendicular to each other and have a magnitude of 1.

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