# Representing a matrix as a product of two matrices in MATLAB

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rihab on 17 Nov 2015
Commented: Torsten on 19 Nov 2015
I have a 4x4 matrix of complex numbers, say
X = [0.4079 + 0.0000i 0.7532 + 0.0030i 0.9791 + 0.0272i 0.9335 - 0.0036i;
0.7532 - 0.0030i 1.2288 + 0.0000i 1.3074 + 0.0052i 0.9791 + 0.0272i;
0.9791 - 0.0272i 1.3074 - 0.0052i 1.2288 + 0.0000i 0.7532 + 0.0030i;
0.9335 + 0.0036i 0.9791 - 0.0272i 0.7532 - 0.0030i 0.4079 + 0.0000i]
I want to represent it as a product of a 4x1 & 1x4 vectors say x such that X = xx^H where H denotes hermitian transpose of x. does anyone have any hint/suggestion to solve this matrix decomposition problem? Any suggestion would be appreciated.
rihab on 18 Nov 2015
@stefan the matrix X is a solution to a least squares problem (AX-B). X is a 16x1 vector and for my purpose I have reshaped into a 4x4 matrix and now I want to decompose this matrix X as a product of two vectors such that X = xx^H (I am basically interested in extracting the four elements of vector x). Since X a rank 4 matrix and cant be decomposed in such a way, I am looking for some approximate solution. Any suggestions on obtaining this vector x would be appreciated.

John D'Errico on 18 Nov 2015
Edited: John D'Errico on 18 Nov 2015
I'm sorry, but this is flat out impossible.
X = [0.4079 + 0.0000i 0.7532 + 0.0030i 0.9791 + 0.0272i 0.9335 - 0.0036i;
0.7532 - 0.0030i 1.2288 + 0.0000i 1.3074 + 0.0052i 0.9791 + 0.0272i;
0.9791 - 0.0272i 1.3074 - 0.0052i 1.2288 + 0.0000i 0.7532 + 0.0030i;
0.9335 + 0.0036i 0.9791 - 0.0272i 0.7532 - 0.0030i 0.4079 + 0.0000i];
Not even close to being possible. Lets see why.
rank(X)
ans =
4
svd(X)
ans =
3.7714
0.62027
0.11221
0.010048
Only if the rank of X was 1, i.e., it had 1 non-zero singular values, and 3 essentially zero values, could you do this.
The point is, to represent the matrix as an outer product of two vectors, i.e., x'*y, where x and y are row vectors, the result would have rank 1. That is a fundamental of linear algebra.
As you can see, that is clearly not true. Simply wanting to do the impossible is not an option. Case closed, IF you want an exact solution.
If you wish to find the closest approximation as such a result (based on reading your comments) then it is possible. Use the SVD.
[U,S,V] = svd(X);
u = U(:,1)
u =
-0.41048 - 4.0811e-18i
-0.57576 + 0.001811i
-0.57571 + 0.0081111i
-0.41042 + 0.0070737i
s = S(1,1)
s =
3.7714
v = V(:,1)
v =
-0.41048 + 0i
-0.57576 + 0.001811i
-0.57571 + 0.0081111i
-0.41042 + 0.0070737i
Since X is symmetric, we see that u and v will be the same. If the goal is to write Xhat as a product w*w', then wejust use
w = u*sqrt(s)
w =
-0.79716 - 7.9255e-18i
-1.1181 + 0.003517i
-1.118 + 0.015752i
-0.79705 + 0.013737i
w*w'
ans =
0.63547 + 0i 0.89134 + 0.0028036i 0.89125 + 0.012557i 0.63538 + 0.010951i
0.89134 - 0.0028036i 1.2502 + 0i 1.2502 + 0.013681i 0.89125 + 0.012557i
0.89125 - 0.012557i 1.2502 - 0.013681i 1.2502 + 0i 0.89134 + 0.0028036i
0.63538 - 0.010951i 0.89125 - 0.012557i 0.89134 - 0.0028036i 0.63547 + 0i
As you can see, this must be rank 1.
rank(w*w')
ans =
1
The norm of the error of approximation is as small as possible.
norm(w*w' - X)
ans =
0.62027
So this is the best way (least squares) to produce a rank 1 approximation to X. By way of comparison,
norm(X)
ans =
3.7714
##### 2 CommentsShowHide 1 older comment
John D'Errico on 18 Nov 2015
Fixed

Torsten on 18 Nov 2015
Edited: Walter Roberson on 18 Nov 2015
Consequently, u1*sigma1*u1^H where u1 is the eigenvector corresponding to the largest eigenvalue sigma1, is the best rank-1 - approximation to X in the Frobenius norm.
Best wishes
Torsten.
##### 2 CommentsShowHide 1 older comment
Torsten on 19 Nov 2015
For Hermitian matrices, the sigma's are eigenvalues, too.
Best wishes
Torsten.