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after a series of computations, I was able to get the following matrix equation from the given of a problem:

[tex]\[\left( \begin{array} {ccc} W_1 \\ W_2 \end{array} \right)\] =

\[\left( \begin{array} {ccc} \frac{\sigma_{11}}{\sqrt{\sigma_{11}^2 + \sigma_{12}^2}} & \frac{\sigma_{12}}{\sqrt{\sigma_{11}^2 + \sigma_{12}^2}} \\ \frac{\sigma_{21}}{\sqrt{\sigma_{21}^2 + \sigma_{22}^2}} & \frac{\sigma_{22}}{\sqrt{\sigma_{21}^2 + \sigma_{22}^2}}\end{array} \right)\] \[\left( \begin{array} {ccc} Y_1 \\ Y_2 \end{array} \right)\] [/tex]

where Y1 and Y2 are independent processes.

the correlation of W1 and W2 was given as follows:

[tex]\rho = \frac{\sigma_{11}\sigma_{21} + \sigma_{12}\sigma_{22}}{\sqrt{\sigma_{11}^2 + \sigma_{12}^2}\sqrt{\sigma_{21}^2 + \sigma_{22}^2}}[/tex]

Now what the problem asks is that I be able to show the following matrix equation to be true:

[tex]\[\left( \begin{array} {ccc} W_1 \\ W_2 \end{array} \right)\] =

\[\left( \begin{array} {ccc} 1 & 0 \\ \rho & \sqrt{1 - \rho^2} \end{array} \right)\] \[\left( \begin{array} {ccc} Z_1 \\ Z_2 \end{array} \right)\] [/tex]

where Z1 and Z2 are independent

and rho is the correlation of W1 and W2.

My question is:

can I just let

[tex]\sigma_{11} = 1[/tex]

[tex]\sigma_{12} = 0[/tex]

and just let

Y1 = Z1

Y2 = Z2?

cause if I did so, then the matrix equation that I want to prove is satisfied.

that is, the following are now true:

[tex]\rho = \frac{\sigma_{21}}{\sqrt{\sigma_{21}^2 + \sigma_{22}^2}}[/tex]

[tex]\sqrt{1 - \rho^2} = \frac{\sigma_{22}}{\sqrt{\sigma_{21}^2 + \sigma_{22}^2}}[/tex]

Is this a valid way of proving?

or should I have to find a matrix linear transformation to transfrom the first equation that I got into the required equation? cause if that's the way that it shouldbe done, then I'm not sure how to proceed about it.

thanks for the help.

[tex]\[\left( \begin{array} {ccc} W_1 \\ W_2 \end{array} \right)\] =

\[\left( \begin{array} {ccc} \frac{\sigma_{11}}{\sqrt{\sigma_{11}^2 + \sigma_{12}^2}} & \frac{\sigma_{12}}{\sqrt{\sigma_{11}^2 + \sigma_{12}^2}} \\ \frac{\sigma_{21}}{\sqrt{\sigma_{21}^2 + \sigma_{22}^2}} & \frac{\sigma_{22}}{\sqrt{\sigma_{21}^2 + \sigma_{22}^2}}\end{array} \right)\] \[\left( \begin{array} {ccc} Y_1 \\ Y_2 \end{array} \right)\] [/tex]

where Y1 and Y2 are independent processes.

the correlation of W1 and W2 was given as follows:

[tex]\rho = \frac{\sigma_{11}\sigma_{21} + \sigma_{12}\sigma_{22}}{\sqrt{\sigma_{11}^2 + \sigma_{12}^2}\sqrt{\sigma_{21}^2 + \sigma_{22}^2}}[/tex]

Now what the problem asks is that I be able to show the following matrix equation to be true:

[tex]\[\left( \begin{array} {ccc} W_1 \\ W_2 \end{array} \right)\] =

\[\left( \begin{array} {ccc} 1 & 0 \\ \rho & \sqrt{1 - \rho^2} \end{array} \right)\] \[\left( \begin{array} {ccc} Z_1 \\ Z_2 \end{array} \right)\] [/tex]

where Z1 and Z2 are independent

and rho is the correlation of W1 and W2.

My question is:

can I just let

[tex]\sigma_{11} = 1[/tex]

[tex]\sigma_{12} = 0[/tex]

and just let

Y1 = Z1

Y2 = Z2?

cause if I did so, then the matrix equation that I want to prove is satisfied.

that is, the following are now true:

[tex]\rho = \frac{\sigma_{21}}{\sqrt{\sigma_{21}^2 + \sigma_{22}^2}}[/tex]

[tex]\sqrt{1 - \rho^2} = \frac{\sigma_{22}}{\sqrt{\sigma_{21}^2 + \sigma_{22}^2}}[/tex]

Is this a valid way of proving?

or should I have to find a matrix linear transformation to transfrom the first equation that I got into the required equation? cause if that's the way that it shouldbe done, then I'm not sure how to proceed about it.

thanks for the help.

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