# Valid covariance matrix?

I am trying to replicate some results from this well-cited paper evaluating various methods to determine the rank of a matrix. They give several covariance matrices, and then sample from a Gaussian distribution with each of these matrices.

However, I get strange results when I try to use one of these covariance matrices. In particular, I am trying to sample from a multivariate Gaussian with the first matrix in Figure 1, but when I do so, every variable is exactly the same!

Here is Matlab code:

c = 0.8;
C = [c c c c 0 0 0 0 0;
c c c c 0  0 0 0 0;
c c c c 0  0 0 0 0;
c c c c 0  0 0 0 0;
0 0 0 0 c  c c 0 0;
0 0 0 0 c  c c 0 0;
0 0 0 0 c  c c 0 0;
0 0 0 0 0  0 0 c c;
0 0 0 0 0  0 0 c c;]
X = mvnrnd(zeros(1,9), C,10)


Which gives

X =

1.0310    1.0310    1.0310    1.0310    0.1740    0.1740    0.1740    1.2928    1.2928
0.3512    0.3512    0.3512    0.3512   -2.0864   -2.0864   -2.0864    0.0000    0.0000
0.7149    0.7149    0.7149    0.7149   -0.5316   -0.5316   -0.5316    2.5462    2.5462
-1.7125   -1.7125   -1.7125   -1.7125    0.9807    0.9807    0.9807   -0.3451   -0.3451
-0.5237   -0.5237   -0.5237   -0.5237   -0.1748   -0.1748   -0.1748    0.1343    0.1343
0.0530    0.0530    0.0530    0.0530   -1.7133   -1.7133   -1.7133   -0.2837   -0.2837
-0.5293   -0.5293   -0.5293   -0.5293   -0.6427   -0.6427   -0.6427   -0.1456   -0.1456
-2.1785   -2.1785   -2.1785   -2.1785   -1.9042   -1.9042   -1.9042    0.7329    0.7329
1.6811    1.6811    1.6811    1.6811    2.4141    2.4141    2.4141    0.0731    0.0731
-0.1646   -0.1646   -0.1646   -0.1646    1.2982    1.2982    1.2982    1.7498    1.7498


Why is every variable the same? Note that if I add the identity matrix to $$C$$, this is no longer the case. But I don't think the problem is that $$C$$ is not positive definite; all of its eigenvalues are non-negative.

• Please explain the sense in which "every variable is the same:" I see three distinct column values in $X,$ exactly as would be expected from the covariance matrix. – whuber Jan 13 at 22:52
• @whuber My bad...I meant within each block. For example, why are the first three columns exactly the same? Shouldn't they be correlated but not exactly the same? I suppose I am expecting this because the covariance is 0.8 and not 1, but perhaps I am confusing it with correlation. How could you tell from that covariance matrix that the variables in each group would be exactly the same? – The_Anomaly Jan 13 at 23:02
• You seem to confusing correlation with covariance. The correlation matrices for each block are all $1$s: everything must be perfectly correlated, whence all values within a block must be constant positive multiples of each other. Because all covariances are the same, that positive multiple must be $1.$ – whuber Jan 13 at 23:05