I am given $3$ normal random variables $X_1$,$X_2$,$X_3$ which are linear combinations of $Z_1$,$Z_2$,$Z_3$. $Z_1$,$Z_2$,$Z_3$ are mutually independent standard normal variables.
I am given a correlation matrix
$X=\begin{bmatrix} 1 & 0.3 & 0.4 \\0.3 & 1 & 0.5 \\0.4 & 0.5 & 1 \end{bmatrix}$
I have to find $M$ which is $3 \times 3$ such that
$\begin{bmatrix} X_1 \\X_2 \\X_3 \\ \end{bmatrix} =M \cdot\begin{bmatrix} Z_1 \\Z_2 \\Z_3 \\ \end{bmatrix}$
Any hints on how to approach this, based on my research , I am getting a stong correlation of google hits on multivariate normal distribution and cholesky method, but I am still not able to relate how cholesky method will help if it is M would be relevant.
octave-3.2.4.exe:10> Q3
Q3 =
1.00000 0.30000 0.40000
0.30000 1.00000 0.50000
0.40000 0.50000 1.00000
octave-3.2.4.exe:11> chol(Q3)
ans =
1.00000 0.30000 0.40000
0.00000 0.95394 0.39835
0.00000 0.00000 0.82542
Thanks for your help!!