This post concerns a question regarding nonnegative matrices, i.e., matrices with all entries nonnegative:
For two nonnegative matrices , if , i.e., is nonnegative as well, is there any relation with their singular values?
As we shall see, indeed, the largest singular value of , denoted as , is larger than the largest singular value of , :
Let us first consider a simpler case when are symmetric, so that , . Here for any symmetric matrix , we denote its eigenvalues as .
To prove the lemma, we first recall the Perron-Frobenius theorem which states that the largest eigenvalue (in magnitude) of a
nonnegative matrix is nonnegative and the eigenvalue admits an eigenvector which is entrywise nonnegative as well.
Using this theorem, we can pick a nonnegative unit norm eigenvector corresponding to the eigenvalue , which is both nonnegative and largest in magnitude. Next, by multiplying left and right of by and respectively, we have
Here step is because is nonnegative and is nonnegative.
The step is because has unit norm and . The step is because . The step is because are symmetric and both are nonnegative so largest eigenvalue is indeed just the singular value due to Perron-Frobenius theorem.
To prove the general rectangular case, we use a dilation argument.
Consider the symmetric matrices and in :
Note that has the largest singular value as and has the largest singular value as .
Now since , we also have . Using Lemma 1, we prove the theorem.
How about the second singular value of and ? We don’t have in this case by considering