Any matrix as product of elementary matrices

[Castilla]

Good day. My question is this.

Let A be a square matrix. I know that if Det (A) is not 0, then A can be put as the product of k elementary matrices.

But in Marsden's Elementary Classical Analysis I have read that ANY matrix can be put as the product of elementary matrices.

Iit is ok, or it is a errata?

Thanks for your answer.

 

[StatusX]

It would depend on how you define "elementary matrices," but if you use the usual definition that they are the matrices corresponding to row transpositions, multiplying a row by a constant, and adding one row to another, it isn't hard to show all such matrices have nonzero determinants, and so by the product rule for determinants, (det(AB)=det(A)det(B) ), the product of elementary matrices must be non-singular.

 

[nocturnal]

If the book states that even noninvertible square matrices can be written as a product of elementary matrices, then that is an error. A square matrix is invertible iff it can be written as a product of elementary matrices.

 

[mathwonk]

you need some hypotheses on the ring of entries in the matrices, e.g. they need to come from a euclidean domain, such as a field or the integers. it is unknown if this is possible for entries from any principal ideal domain i believe.

 

[DeadWolfe]

I would think, seeing as the book is a book on classical analysis, that it's safe to assume the Matrices have entries in R.

 

[mathwonk]

just trying to teach you something, which of course you are free to ignore. actually clasical analysis was usually done over C.

 

[Castilla]

Thanks for your answers.

Castilla.

 

[mathwonk]

perhaps you are satisfied, but i am still interested in this topic as i have to teach it soon.

i believe if you think abut it you will notice that the key to writing a matrix as a product of elementary ones, is diagonalizing it by elementary operations.

if the diagonl version is the dientity, then the matrix it self is the product of the elementary matrices corresponding to the elementary operations.

the key to diagonalizing a matrix, is the proces of replacing an element by the gcd of that element and another element in the matrix. thus to diagonalize a matrix roughly this way, one needs to be able to write the gcd of two elements as a linear combination of the two elements.

this is only possible in a p.i.d. But the ooperations called elementary are more restrictive than this. if you think about it, you will notice that the elmentary matrix operations only allow one to replace an element by a unit times itself, plus any multiple of another element.

this is no restriction for fields since all non zero elements are units, but for rings it is a restriction. however in a euclidean domain such as Z, the gcd of a,b, can be obtained by the euclidean algorithm as a combination of elements of form a + by, rather than ax+by.

This makes it possible always to diagonalize any matrix, not just an invertible one, by elementary row and column ooperations over any p.i.d. perhaps this is what marsden was thinking of.

But the diagonal version will not be the identity in case the matrix is not invertible, so one cannot get the original matrix as a product of elementary ones, rather one gets the original matrix as a product of elementary matrices times a diagonal matrix.

does that help?

 

[mathwonk]

by the way my earlier remark may be wrong. it is possibly known that one cannot write all invertible matrices as a product of elementary ones in a pid, but what is not known may be just which matrices can be so written? anyway these are interesting questions about an elementary subject.

if you are primarily interested in fields such as R or C, note that the same operations work in other fields such as finite fields, or rational functions.

But esentially the same operations also work as noted above in pids such as the ring of polynomials over a field. this simple remark leads to the easiest proof of all the standard canonical forms for matrices with coefficients in a field, such as rational canonical forms, and jordan matrices.

i.e. all that is needed to find canonical form of a square matrix like A, over Q say, is to diagonalize the matrix [A-X.I] over the polynomials with rational coefficients.

so if you really understand the proces over a field, it leads much further thn you might think.