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Can two different matrices have the same covariance?

Let's using MATLAB's function cov to compute the covariance of a matrix A and a matrix B. A and B are different, but could them have the same covariance in practice?

Normally, they don't, but are there scenarios when they do?

euraad
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2 Answers2

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Yes, for example you can swap the order of the observations that will not change the final covariance.

Lelouch
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  • Thank you for your answer. Can you show me an example? – euraad Jan 15 '22 at 16:57
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    Also you can add a constant to all of the observations and it won't change the covariance – Ofek Gillon Jan 15 '22 at 17:00
  • @OfekGillon Uh? You mean that cov(A*5) == cov(A) ? – euraad Jan 15 '22 at 17:02
  • I guess its rather Cov(A+5) – Lelouch Jan 15 '22 at 17:03
  • @Lelouch I understand now! Thank you – euraad Jan 15 '22 at 17:04
  • @Lelouch But swapping the order will change the covariance matrix, no? Consider swapping the 1st and 2nd entry of a random vector $X=(X_1,...,X_n)'$. Then the (1,1) entry of the covariance matrix changes from $Var(X_1)$ to $Var(X_2)$. – Golden_Ratio Jan 15 '22 at 17:06
  • @Golden_Ratio I assumed that the author was talking about calculating the covariance matrix from an empirical observation matrix (as is often the case on Matlab). It is the realizations of $X_i$ that I propose to swap, not the random variables themselves, as you usually use multiples observations from the same random vector to compute the empirical covariance matrix. – Lelouch Jan 15 '22 at 17:32
  • @Lelouch Ah makes sense, thanks! – Golden_Ratio Jan 15 '22 at 20:56
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The covariance matrix is a generalization of the scalar notion of variance. Just as different random variables can have the same variance, it is possible different random vectors can have the same covariance matrix.

For instance, consider $X\sim N(0_{2\times 1},\mathbb{I}_2),Y\sim N((1,1)',\mathbb{I}_2).$

Golden_Ratio
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