LU Factorization

[mammothrob]

Im doing a LU factorization on Matrix A

\(\displaystyle A = \left[ {\begin{array}
1 & 1 & 1 \\
1 & 2 & 2 \\
1 & 2 & 3 \\
\end{array}} \right]\)

I reduced it to an upper triangular matricie

\(\displaystyle \left[ {\begin{array}
1 & 1 & 1 \\
0 & 1 & 2 \\
0 & 0 & -1 \\
\end{array}} \right]\)

Along the way I kept track of the 3 elementry inverse matrices that got me there. I multiplied them togeather and got

\(\displaystyle E1*E2*E3*= \left[ {\begin{array}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 1 & 0 \\
\end{array}} \right]\)

which is not a lower triangular matricie, but when I multiply it with the upper triangle I found I get A.



\(\displaystyle \left[ {\begin{array}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 1 & 0 \\
\end{array}} \right]\left[ {\begin{array}
1 & 1 & 1 \\
0 & 1 & 2 \\
0 & 0 & { - 1} \\
\end{array}} \right] = A\)

Does this count as an LU factorization? or just some other factoriztion that I worked out?

If not, how do I make my (inverse elementry matricies) E1E2E3 = Lower triangular?

Any ideas are much appreciated.

 

[Unco]

Firstly, check your three elementary row operations. The rest of your post seems confused.

The algebra is not ambiguous:

\(\displaystyle E_3E_2E_1A = U \, \Rightarrow \, A = E_1^{-1}E_2^{-1}E_3^{-1}U = LU\)

where L is in lower triangular form since there were no row interchanges, and U is in upper triangular form.

 

[JakeD]

Im doing a LU factorization on Matrix A

\(\displaystyle A = \left[ {\begin{array}
1 & 1 & 1 \\
1 & 2 & 2 \\
1 & 2 & 3 \\
\end{array}} \right]\)

I reduced it to an upper triangular matricie

\(\displaystyle \left[ {\begin{array}
1 & 1 & 1 \\
0 & 1 & 2 \\
0 & 0 & -1 \\
\end{array}} \right]\)

Along the way I kept track of the 3 elementry inverse matrices that got me there. I multiplied them togeather and got

\(\displaystyle E1*E2*E3*= \left[ {\begin{array}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 1 & 0 \\
\end{array}} \right]\)

which is not a lower triangular matricie, but when I multiply it with the upper triangle I found I get A.



\(\displaystyle \left[ {\begin{array}
1 & 0 & 0 \\
1 & 1 & 1 \\
1 & 1 & 0 \\
\end{array}} \right]\left[ {\begin{array}
1 & 1 & 1 \\
0 & 1 & 2 \\
0 & 0 & { - 1} \\
\end{array}} \right] = A\)

Does this count as an LU factorization? or just some other factoriztion that I worked out?

If not, how do I make my (inverse elementry matricies) E1E2E3 = Lower triangular?

Any ideas are much appreciated.

I calculated \(\displaystyle U =
\left[ {\begin{array}
1 & 1 & 1 \\
0 & 1 & 1 \\
0 & 0 & 1 \\
\end{array}} \right]\)

and \(\displaystyle L =
E_1^{-1} E_2^{-1} E_3^{-1} =

\left[ {\begin{array}
1 & 0 & 0 \\
1 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]

\left[ {\begin{array}
1 & 0 & 0 \\
0 & 1 & 0 \\
1 & 0 & 1 \\
\end{array}} \right]

\left[ {\begin{array}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 1 & 1 \\
\end{array}} \right]

=
\left[ {\begin{array}
1 & 0 & 0 \\
1 & 1 & 0 \\
1 & 1 & 1 \\
\end{array}} \right]\)

which multiply to A.