the SVD provides real-valued rank

Suppose that X has a singular value decomposition 

X = u\ w\ v^T ;

n nonzero singular values (nonzero elements of w);

the smallest nonzero singular value is \epsilon > 0 .


X is of rank n

X differs from a matrix Y of rank n-1 by \epsilon ;

no matrix of rank n-1 differs from X by less than \epsilon

we may compute one such matrix Y as follows: 

the matrix

Y = u\ w2\ v^T

is of rank n-1 and differs from X by \epsilon  , where w2 is w with \epsilon  replaced by 0.

If, for example, X had 5 nonzero singular values and the smallest one was .15, then we could say that the rank of X was 4.15 . and if we replace the .15 in w by 0 in w2 and compute Y, we get a matrix Y of rank 4 which differs from X by as little as possible.
i remark that i do this by changing a value in the matrix w, leaving u and v alone; hence Y is the same shape as X. There are those who would drop a row and column from w, then drop a column from each of u and v. (see below under alternative SVD.)


Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s

%d bloggers like this: