Preserving Rank: Exploring Elementary Operations in Matrix Algebra

[swampwiz]

What is the rule on the preservation of rank through an elementary operation? I know that rank can never go up, but is there any direct way to determine that it goes down (either than reducing the matrix down to row-echelon form)? Is there a good source that go into the proofs for properties of rank like this?

A side question is is it possible to have a linear dependency on rows as well as columns, or is the whole notion of such a dependency moot as both types are fungible?

Thanks

 

[chiro]

Hey swampwiz.

You might want to study how rank changes as a function of operators with certain rank.

I think you should find that it will be a minimum of both "ranks" and you can look at how this is done by expanding the matrix multiplication as a function of basis vectors.

Remember that if you get collinearity, then you will be able to group two vectors together.

As an example, if you have ax + by + cz [example with rank three] and x and y are collinear then you can re-write that as (a+bd)x + cz which reduces this to rank 2.

 

[swampwiz]

You didn't seem to answer my question, but I think I have figured it out on my own.

Any elemental operation that has an inverse must preserve rank, and so the only such operation that does not have an inverse is the scale operation with the scale factor being 0, which ends up zeroing out that row or column, and if that row or column had not been a zero vector, then the number of rows or columns will go down by 1, which would decrease the rank if the original number of non-zero rows or columns were not greater than the non-zero columns or rows, respectively.

 

[Erland]

What is an "elementary operation"?

An elementary row operation on a matrix is usually defined as an operation of one the following three types:

1. Multiplication of a row by a nonzero constant.
2. Addition by a multiple of a row to another row.
3. Interchanging two rows with each other.

As you noted, an elementary row operation does not change the rank of a matrix. But you also seems to have another opertion in mind, namely multiplying a row by zero, making it into a zero row. In this case, the rank may decrease by one (not with more) but not always. If the row is already a zero row, or more generally, if it can be written as a linear comination of the other rows, the dimension of the row space, i.e. the rank, does not change after such an operation.

 

[mathwonk]

you may enjoy showing that each (invertible) elementary operation can be achieved by multiplication by an invertible matrix, and this is another way to see it does not change rank. yes there are also column operations parallel to row operatioons and they are also invertible, and preserve rank (row rank = column rank).