TDSM 8.9

LU factorization of a matrix is not necessarily unique. Example: proof for $$2 \times 2$$ square matrix:

Let $$L = \begin{bmatrix} 1 & 0\\ l & 1 \end{bmatrix}$$ , $$U = \begin{bmatrix} u_1 & u_3\\ 0 & u_2 \end{bmatrix}$$

$$\Rightarrow LU = \begin{bmatrix} u_1 & u_3\\ lu_1 & lu_3+u_2 \end{bmatrix}$$

Let $$M = \begin{bmatrix} m_1 & m_3\\ m_2 & m_4 \end{bmatrix} = LU$$

$$\Rightarrow \left\{ \begin{array}{lr} u_1 = m_1 & \\ lu_1 = m_2 & \\ u_3 = m_3& \\ lu_3 + u_2 = m_4 \end{array} \right.$$

Let $$m_1 = m_2 = 0 \Rightarrow$$ There are 3 equations for 4 variables $$\Rightarrow$$ There are many value for $$l$$ satisfies the equations.

$$\Rightarrow$$ LU factorization of $$M$$ not unique.