Proof: Matrix Rank of X = n with Y,Z of Rank n-1,1 Respectively

[rhuelu]

I'm trying to show that any matrix X with rank n can be written as the sum of matrices Z and Y with rank n-1 and 1, respectively.

Since X,Y, Z have the same dimensions, is this a simple matter of saying pick one of the columns in X with a pivot. Let Z= X with this column replaced by zeroes but all other entries the same as X. Let Y consist of the removed column and all other entries 0. Thus, X=Y+Z and Y has rank n-1 and Z has rank 1.

Does this look correct?

 

[morphism]

Yes, that's fine.

 

[rhuelu]

it doesn't seem like this is the most formal logic in the world...is it trivial that the matrices constructed have ranks of n-1 and 1 or does this need to be shown as well

 

[morphism]

Since the rank of X is n, its columns are linearly independent. Hence any subset of those columns is linearly independent as well.