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

[Clandry]

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?

 

[Mark44]

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, S^T? Isn't that orthogonal as well?

 

[Clandry]

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, S^T? Isn't that orthogonal as well?

Ah yes! I think I overthought this problem.