Recover matrix Z from XZX', Z es symmetric and n-by-n, while X is k-by-n where n>>k

I have the matrices (XZX') and X and I want to recover Z. Dimension: X is kxn, Z is nxn, and n >> k. I know that Z is simetric

[David Goodmanson]

Hi Patrcio,

this can't be done in any unique way. For example, take the simple case where X is a 1xn row vector and Z is diagonal. Then the equation reduces to Sum X_i*Z_ii*X_i = A where A is a scalar. That's one equation, n unknowns.

[PATRICIO AROCA]

Thanks David for your answer!!!

Let say that I know M = XZX' is a variance-covariance matrix that I have and it is 5x5. I also know X which is a 27x5 and I know that M is simetryc and it was built multypling XZX'. I thought that there was no way to recover Z, but I give a shot that someone might face and solve this ill-problem before. In addition, in only very special case Z is diagonal, in most cases it is not.

[Matt J]

Let say that I know M = XZX' is a variance-covariance matrix that I have and it is 5x5. I also know X which is a 27x5

If X is 27x5, then M must be 27x27, not 5x5. Do you mean that Z is 5x5? If so, then you really should have written k>>n in your initial post.

[PATRICIO AROCA]

Sorry, my mistake. M is 5x5, and X is 5x27

[David Goodmanson]

Hi Patricio, so now Z is 27x27 which is back to an underdetermined solution for Z.

Answer 1

[Matt J]

In the case where k truly is <<n, you can use my KronProd class to get the minimum norm solution

https://www.mathworks.com/matlabcentral/fileexchange/25969-efficient-object-oriented-kronecker-product-manipulation

k=10; n=100;

X=rand(k,n);
Ztrue=rand(n); Z=Z+Z.';

M=X*Ztrue*X.';

K=KronProd({X,X});
tic;
Z =  pinv(K)*M ;
toc; %Elapsed time is 0.005358 seconds.

Naturally, you should not expect the result to equal the under-determined Ztrue.

[David Goodmanson]

Hi Matt, I tried to reply to your last comment but that answer is gone. I indeed did not see your updated answer when I posted my answer. Sorry I assumed wrongly, it makes sense now.

[Matt J]

No worries - the hazards of browser delays and such...

Answer 2

[KSSV]

Does this match your criteria?

k = 8 ; n = 5 ; 
% create dummy data 
X = rand(n,k) ; 
Z = rand(n) ; 
Z = Z+Z' ;    % make Z symmetric 
D = X'*Z*X ;  % known value 

%% solve for Z knowing D and X
Z0 = inv(X*X')*X*D*X'*inv(X*X')  ;   % this is same as Z

Answer 3

[Bruno Luong]

You can't getback to 27x27 covariance matrix Z after reducing it it on 5 dimensional space (by X). The information lost forevver.