# The decomposition for a symmetric positiv definite matrix is unique

We have the matrix $$\begin{equation*}A=\begin{pmatrix}1/2 & 1/5 & 1/10 & 1/17 \\ 1/5 & 1/2 & 1/5 & 1/10 \\ 1/10 & 1/5 & 1/2 & 1/5 \\ 1/17 & 1/10 & 1/5 & 1/10\end{pmatrix}\end{equation*}$$

I have applied the Cholesky decomposition and found that $$A=\tilde{L}\cdot \tilde{L}^T$$ where
$$\begin{equation*}\tilde{L}=\begin{pmatrix}\frac{1}{\sqrt{2}} & 0 & 0 & 0 \\ \frac{\sqrt{2}}{5} & \sqrt{\frac{21}{50}} & 0 & 0 \\ \frac{\sqrt{2}}{10} & \frac{4\sqrt{42}}{105} & \frac{2\sqrt{1155}}{105} & 0 \\ \frac{\sqrt{2}}{17} & \frac{13}{17\sqrt{42}} & \frac{142}{17\sqrt{1155}} & \sqrt{\frac{298}{15895}}\end{pmatrix} \ \ \ \text{ und } \ \ \ \tilde{L}^T=\begin{pmatrix}\frac{1}{\sqrt{2}} & \frac{\sqrt{2}}{5} & \frac{\sqrt{2}}{10} & \frac{\sqrt{2}}{17} \\ 0 & \sqrt{\frac{21}{50}} & \frac{4\sqrt{42}}{105} & \frac{13}{17\sqrt{42}} \\ 0 & 0 & \frac{2\sqrt{1155}}{105} & \frac{142}{17\sqrt{1155}} \\ 0 & 0 & 0 & \sqrt{\frac{298}{15895}} \end{pmatrix}\end{equation*}$$

Is that correct?

I want to show that the decomposition for a symmetric positiv definite matrix is unique and it is given as a hint that we have to use the LU decomposition.

Could you explain to me how we can do that?