# How to generate a multivariate normal distribution by conditioning on the normally distributed mean of each sample?

I would like to generate a joint distribution of random variables $$X$$ and $$Y$$ from a multivariate normal distribution with mean zero and symmetric covariance matrix. However, each generated $$x_i$$ and $$y_i$$ needs to be constrained by a mean, $$z_i$$, that was previously generated from a normal distribution with the same variance as $$X$$ and $$Y$$. For each sample generated, how do I condition on $$z_i$$ to sample $$x_i$$ and $$y_i$$, while maintaining the desired covariance matrix for the joint distribution of $$X$$ and $$Y$$?

Here is an example: Let's say I ultimately want a multivariate normal distribution of $$X$$ and $$Y$$ with: $$\mu = \left(\matrix{0 \\ 0}\right)$$ and $$\Sigma = \left(\matrix{\sigma_{11} & \sigma_{12} \\ \sigma_{21} & \sigma_{22}}\right)$$, where $$\sigma_{11}=\sigma_{22}$$. I have sampled $$Z$$ from normal distribution: $$Z \sim \mathcal{N}(0,\sigma_{11})$$. For each $$z_i$$, I want to sample $$x_i$$ and $$y_i$$ such that they have mean $$z_i$$, but overall will have a covariance matrix $$\Sigma$$ just described.

I think I can sample each $$x_i$$ and $$y_i$$ independently from a normal distribution with mean $$z_i$$, but I'm not sure what the variance should be. I have found a distribution to sample one value conditional on knowing the other that it covaries with, but I believe that's different from what I'm trying to do here. Thank you in advance for helping me identify the right distribution around each $$z_i$$!

If$$\left(\matrix{X \\ Y}\right)\sim \mathcal N \left(\mu,\Sigma\right)$$then $$X\sim\mathcal N(0,\sigma_{11})$$and$$Y|X\sim\mathcal N(\sigma_{12} X/\sigma_{11},\sigma_{22}-\sigma_{11}^2/\sigma_{11}^2)$$ meaning that $$Z$$ and $$X$$ have the same distribution.