MHB New to Linear Algebra - LU Decomposition

[pp123123]

Just came across LU decomposition and I am not sure how to work on this problem:

Let L and L₁ be invertible lower triangular matrices, and let U and U₁ be invertible upper triangular matrices. Show that LU=L₁U₁ if and only if there exists an invertible diagonal matrix D such that L₁=LD and U₁=D^-1U. [Hint: Scrutinize L^-1L₁=UU₁^-1]

I could work on the part till L^-1L₁=UU₁^-1, but I am not sure what I could do further. Give me some hints (and I don't actually know how to prove iff statements)?

Thankss!

 

[Opalg]

Just came across LU decomposition and I am not sure how to work on this problem:

Let L and L₁ be invertible lower triangular matrices, and let U and U₁ be invertible upper triangular matrices. Show that LU=L₁U₁ if and only if there exists an invertible diagonal matrix D such that L₁=LD and U₁=D^-1U. [Hint: Scrutinize L^-1L₁=UU₁^-1]

I could work on the part till L^-1L₁=UU₁^-1, but I am not sure what I could do further. Give me some hints (and I don't actually know how to prove iff statements)?

In the equation $L^{-1}L_1 = UU_1^{-1}$, the left side is a lower-triangular matrix, and the right side is an upper-triangular matrix. If they are equal then they must represent a matrix that is both lower-triangular and upper-triangular. What can you say about such a matrix?

To prove an iff statement, you must show that the implication works in both directions. In this case, you first need to prove that if $LU = L_1U_1$ then there exists an invertible diagonal matrix $D$ such that $L_1 = LD$ and $U_1 = D^{-1}U$. Then you also have to prove the converse implication, namely that if there exists an invertible diagonal matrix $D$ such that $L_1 = LD$ and $U_1 = D^{-1}U$ then it follows that $LU = L_1U_1$.

 

[pp123123]

In the equation $L^{-1}L_1 = UU_1^{-1}$, the left side is a lower-triangular matrix, and the right side is an upper-triangular matrix. If they are equal then they must represent a matrix that is both lower-triangular and upper-triangular. What can you say about such a matrix?

To prove an iff statement, you must show that the implication works in both directions. In this case, you first need to prove that if $LU = L_1U_1$ then there exists an invertible diagonal matrix $D$ such that $L_1 = LD$ and $U_1 = D^{-1}U$. Then you also have to prove the converse implication, namely that if there exists an invertible diagonal matrix $D$ such that $L_1 = LD$ and $U_1 = D^{-1}U$ then it follows that $LU = L_1U_1$.

Oh I get it. So is it okay to write something like:

Due to the fact that $L^{-1}L_1 = UU_1^{-1}$
The resulting matrix must be a diagonal matrix under the circumstances that it must both be a lower-triangular and upper-triangular matrix.
Denote $D$ as the desired diagonal matrix.

$L^{-1}L_1 = UU_1^{-1} = D$
As D is a product of two invertible matrices, D must be invertible as well. Moreover,
$L^{-1}L_1 = D$
$LL^{-1}L_1=LD$
$L_1=LD$

$UU_1^{-1}U_1=DU_1$
$U=DU_1$
$D^{-1}U=D^{-1}DU_1$
$U_1=D^{-1}U$

On the other hand, given $L_1=LD$ and $U_1=D^{-1}U$,
$L_1U_1=LDD^{-1}U$
$L_1U_1=LU$

Thus, the statement is proved.

Much Thanks!