# Additional Property of Singular Value Decomposition

I am new to SVD so forgive me if the question is trivial. Following is my question. If I have two sets of linear equations,

Y1 = T1.X

Y2 = T2.X

where T1 and T2 are mxn rectangular matrices. Now let's assume that both equations have some redundant linear equations, with $$\sigma_1$$ and $$\sigma_2$$ (both less than m, n) non-zero singular values. If I construct a new sets of linear equations by taking sum of the above equations,

Y3 = T3.X

where Y3 = Y1+Y2 and T3 = T1+T2, how does the number of singular value change for the combined linear equation?

Also, assuming the transformation matrices T1 is less noisy than T2, how would the noise property of the new T3 matrix?

• Have you considered constructing small examples? What, for instance, happens to the number of singular values when $T_1=-T_2$? What happens when the upper (or leftmost) square portion of $T_1+T_2$ is the identity matrix? Regardless, concerning your last question, could you please explain what the "noise property" of a matrix might be? – whuber Jan 11 at 17:56
• @whuber, I have not looked tried constructing an example for it (I will do so), the problem sounded "Trivial" to those who has working knowledge on SVD, thus wanted to see if there is some theorem that answers my question. Regarding the noise, I am talking about the size of the singular values. For example, figure 3 of this paper (osapublishing.org/oe/…) plots the author plots SVD values and identify the region where the values are close to zero, thus consider them to be noise, thus my question on how adding impacts such values. – crossingsymmetry Jan 11 at 18:10
• That raises the question of what "close to zero" might mean. It probably depends on the context and sounds subjective. The two kinds of examples I urged you to examine exhibit extreme cases of how adding can change the singular values. – whuber Jan 11 at 18:59
• @whuber That is another question I have. I assume singular values correspond to "weight" of the basis, but how would one simulate a matrix to control the dynamic range of singular values and non-zero fraction of singular values? – crossingsymmetry Jan 11 at 19:44
• Just simulate the $U,D,$ and $V$ vectors in the SVD and multiply them. An easy way to simulate $U$ and $D$ is to generate an $X$ with iid standard Normal coordinates and extract $U$ and $V$ from its SVD. The diagonal of $D$ contains the singular values, which you have to generate yourself. Then remultiply everything to obtain your simulated matrix. – whuber Jan 11 at 20:08