Changing diagonal elements of a matrix

[adelaide_user_1009]

TL;DR

Can I transform only diagonal elements of a variance-covariance matrix?

I have a variance-covariance matrix W with diagonal elements diag(W). I have a vector of weights v. I want to scale W with these weights but only to change the variances and not the covariances. One way would be to make v into a diagonal matrix and (say V) and obtain VW or WV, which changes both diagonal and off-diagonal elements of W. Does it make sense to only multiply diag(W) with v and leave the off-diagonal elements of W untouched?

I have searched for any intuition around it but found nothing that supports it or otherwise. Any help would be appreciated on references, texts, etc that would involve such a situation.

 

[fresh_42]

Why?

The variance is only a specific case of covariance. Hence changing it means changing a covariance automatically.

Assume for a moment that you were successful. What would the new matrix represent? Definitely no covariances anymore.

Assume a very simple case: $X_k \longmapsto X'_k:=\alpha_kX_k.$ Then $\operatorname{cov}(X'_m,X'_n)=\alpha_m\alpha_n \operatorname{cov}(X_m,X_n).$ Since all covariances should remain unchanged, we get $\alpha_m\alpha_n = 1$ for all $m\neq n$. If we have enough indices, then we will be left with $\alpha_k=1$ for all $k.$

You are asking the wrong question.

What do you intend to do, in the sense that your result will still have a meaning?​

 

[Office_Shredder]

A matrix is a covariance matrix if and only if it is symmetric and positive semi definite. So you can change the diagonal as long as you don't make it too small, and it should continue to satisfy these properties. It will be the covariance matrix for a totally different set of random variables though as fresh points out.

As long as you want to increase the diagonal entries, you can just add some new random variable that is uncorrelated with all your existing random variables whose variance is equal to the increase. Making them smaller requires something more clever (and isn't anyways possible)

 

[adelaide_user_1009]

A matrix is a covariance matrix if and only if it is symmetric and positive semi definite. So you can change the diagonal as long as you don't make it too small, and it should continue to satisfy these properties. It will be the covariance matrix for a totally different set of random variables though as fresh points out.

As long as you want to increase the diagonal entries, you can just add some new random variable that is uncorrelated with all your existing random variables whose variance is equal to the increase. Making them smaller requires something more clever (and isn't anyways possible)

You are right — changing only the diagonal of the covariance matrix would not transform the matrix and it would not be the covariance matrix of the variables anymore. I will try to explore the possibility of a convex combination of the full matrix W and matrix with only diag(W). Something like (1-a)diag(W) + aW

 

[Office_Shredder]

That is the covariance matrix of $\sqrt{a}$ multiplied by your original variables, plus new uncorrelated random variables added to each of them with variance $(1-a)$ times the variance of the variable it is being added to.

 

[adelaide_user_1009]

That is the covariance matrix of $\sqrt{a}$ multiplied by your original variables, plus new uncorrelated random variables added to each of them with variance $(1-a)$ times the variance of the variable it is being added to.

Right. So assuming no restriction is imposed on their correlation, $a = 1$ and W is true and assuming variables are totally uncorrelated, $a = 0$ diagonal matrix of diag(W) is true. I need to figure out how to specify $a$ in my case.

 

[Office_Shredder]

I think I don't fully understand when you say you have a vector of weights, how those are supposed to modify the variances