Suppose that X has a singular value decomposition

;

n nonzero singular values (nonzero elements of w);

the smallest nonzero singular value is .

Then,

X is of rank n

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

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

we may compute one such matrix Y as follows:

the matrix

is of rank n-1 and differs from X by , where w2 is w with 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.)

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