# What does covariance matrix in probability density function signify

I referred the literature and understood that in the image shown the sigma square multiplied by Identity matrix represents covariance matrix. But in many cases the distribution is given without identity matrix. So what exactly this identity matrix signify in this context. Expecting a small example that will clarify the difference. Thanks in advance....

• This is a standard notation for a Complex Normal distribution.
– whuber
Commented May 13, 2020 at 12:28
• But i have seen many examples where complex normal distribution is used without identity matrix. Commented May 13, 2020 at 12:33
• That's fine--this is a specific case of that. I can't believe you are trying to ask what the identity matrix is, but I can't fathom what you are looking for in an answer.
– whuber
Commented May 13, 2020 at 14:25

It sounds like maybe you're confused why there's a matrix, rather than a number?

If that's the case, it's just that here you have a multivariate distribution. It is equivalent (in the case where you have a scalar multiple of the identity) to taking each diagonal, and having $$N$$ $$\mathcal{CN}(0,\sigma_0^2)$$ distributions.

So if you sampled from the distribution in your original post, you'd get a vector of length $$N$$, with each element sampled from $$\mathcal{CN}(0,\sigma_0^2)$$.

This becomes more useful when your covariance matrix isn't diagonal. In that case, you can't just draw each element separately, as the correlations between different parameters will be non-zero.

The alternative answer, if you're sure your dealing with multivariate distributions in all cases you've seen, is that the authors have just been lazy and neglected to include the identity matrix. In this case, the only assumption you can make is that each parameter has the same variance (which is represented my multiplying it by an identity matrix).

• Ok....understood Commented May 13, 2020 at 23:33

You use an identity matrix multiplied by some number when you want to say the covariance matrix is diagonal and has the same value in every element of the diagonal.

• If i have 5 numbers that follows circularly symmetric complex gaussian distribution, then can you please explain your answer for this 5 numbers. Because this is where i am getting stuck. Commented May 13, 2020 at 12:33