LU-factorization: why can't I get a unit lower triangular matrix?

Question

I want to find an $LU$-factorization of the following matrix: \begin{align*} A = \begin{pmatrix} 3 & -6 & 9 \\ -2 & 7 & -2 \\ 0 & 1 & 5 \end{pmatrix} \end{align*} This matrix is invertible (the determinant is $33$), so I should be getting a $LU$ decomposition where $L$ is a lower triangular matrix with only $1s$ on the diagonal. But that's not what I'm getting. Here is what I did: \begin{align*} \begin{pmatrix} 3 & -6 & 9 \\ -2 & 7 & -2 \\ 0 & 1 & 5 \end{pmatrix} & \begin{matrix} \xrightarrow{R_1 \rightarrow 1/3 R_1} \end{matrix} \begin{pmatrix} 1 & -2 & 3 \\ -2 & 7 & -2 \\ 0 & 1 & 5 \end{pmatrix} \begin{matrix} \xrightarrow{R_2 \rightarrow R_2 + 2R_1} \end{matrix} \begin{pmatrix} 1 & -2 & 3 \\ 0 & 3 & 4 \\ 0 & 1 & 5 \end{pmatrix} \\ & \begin{matrix} \xrightarrow{R_2 \rightarrow 1/3 R_2} \end{matrix} \begin{pmatrix} 1 & -2 & 3 \\ 0 & 1 & \frac{4}{3} \\ 0 & 1 & 5 \end{pmatrix} \begin{matrix} \xrightarrow{R_3 \rightarrow R_3 - R_2} \end{matrix} \begin{pmatrix} 1 & -2 & 3 \\ 0 & 1 & \frac{4}{3} \\ 0 & 0 & \frac{11}{3} \end{pmatrix} = U. \end{align*} This matrix is now in echelon form. The elementary matrices corresponding to the operations are: \begin{align*} E_1 & = \begin{pmatrix} \frac{1}{3} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \qquad (R_1 \rightarrow 1/3 R_1) \quad E_2 = \begin{pmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \qquad (R_2 \rightarrow R_2 + 2R_1) \\ E_3 &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & \frac{1}{3} & 0 \\ 0 & 0 & 1 \end{pmatrix} \qquad (R_2 \rightarrow 1/3 R_2) \quad E_4 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & -1 & 1 \end{pmatrix} \qquad (R_3 \rightarrow R_3 - R_2) \end{align*} Hence we have $E_4 E_3 E_2 E_1 A = U$. Now \begin{align*} E_4 E_3 E_2 E_1 = M = \begin{pmatrix} \frac{1}{3} & 0 & 0 \\ \frac{2}{9} & \frac{1}{3} & 0 \\ - \frac{2}{9} & - \frac{1}{3} & 1 \end{pmatrix} \end{align*} is a lower triangular matrix. Now we can write \begin{align*} A = (E_4 E_3 E_2 E_1)^{-1} U = M^{-1} U = LU, \end{align*} with \begin{align*} M^{-1} = L = \begin{pmatrix} 3 & 0 & 0 \\ -2 & 3 & 0 \\ 0 & 1 & 1 \end{pmatrix} \end{align*} But why does my matrix $L$ not have $1s$ on the diagonal? I thought an $LU$-factorization of an invertible matrix is unique, and that in that case $L$ is an unit lower triangular matrix? Did I overlook something or made a mistake? I haven't interchanged any rows. Please help!

Answer 1

In the framework of the Gaussian elimination procedure, you get $L$ with all $1$s on the diagonal by never truly rescaling rows. You rescale them in the intermediate step before you add them to another row, but then you "scale them back" so that you're only changing the row that you actually added to. You can always perform Gaussian elimination in this way. (By contrast, you may not always be able to perform Gaussian elimination without exchanging rows to obtain your next pivot, so an $LU$ decomposition may not always exist.)

Answer 2

Uniqueness is only for such $L$ that $L_{ii} = 1$ for all $i$. To achieve this, move the scaling factors to $U$ like this: $$A = LU = (LS)(S^{-1}U) = \tilde L \tilde U$$ Here you need $$S = \pmatrix{\frac13&&\\&\frac13&\\&&1}$$