Prove that the LU factorization is unique by considering the possibility that LU = , for some L, lower triangular matrices, U, upper triangular matrices, such that L or U .
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Cholesky factorization for $2 \times 2$ matrices. Show that any positive definite $2 \times 2$ matrix $A$ can be written uniquely as $A=L L^{T}$, where $L$ is a lower triangular $2 \times 2$ matrix with positive entries on the diagonal. Hint: Solve the equation $$\left[\begin{array}{ll} a & b \\ b & c \end{array}\right]=\left[\begin{array}{ll} x & 0 \\ y & z \end{array}\right]\left[\begin{array}{ll} x & y \\ 0 & z \end{array}\right]$$
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