Can two different matrices have the same covariance?

Question

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?

Answer 1

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

Answer 2

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