Welcome to MathOverflow

Question

0

I'm looking for a reference of the following statement (which can easily be proved by Laplace's formula and induction):

Let $R$ be a commutative ring with identity and let $A$ be an invertible matrix over $R$. Then there is a permutation matrix $P$ such that the diagonal of $PA$ has no zero.

Edit: Of course, $R$ has to be a domain to make the arguments (formula of Leibniz or Laplace) work. Futhermore, it's sufficient to require $det(A) \neq 0$ (instead of $A$ being invertible).

edited May 6 2011 at 18:56

asked May 5 2011 at 21:00

Ralph
12.5k●10●36

I'm hoping you won't find a reference of that statement, since it is false. (Consider the 1x1 matrix over the zero ring.) – Ricky Demer May 5 2011 at 21:13

Doesn't a ring with identity have at least two elements? – Tony Huynh May 5 2011 at 21:19

No, Tony. Otherwise algebra would be swamped in statements like "if $I$ is an ideal of $R$, then $R/I$ is a ring or zero". – darij grinberg May 5 2011 at 21:21

Makes sense. Thanks darji. – Tony Huynh May 5 2011 at 21:26

3

I don't know of a reference, but you can also appeal to the determinant formula as a sum of (permuted) products for a shorter proof. Then you may not need a reference. Gerhard "Ask Me About System Design" Paseman, 2011.05.05 – Gerhard Paseman May 5 2011 at 22:08

show 8 more comments

Denis Serre · Answer 1 · 2011-05-06 07:29:24Z UTC

3

There is a strictly stronger fact about matrices:

Given $A\in{\bf M}_n(R)$, the following statements are equivalent.

There exists a permutation matrix $P$ such that the diagonal of $PA$ has no zero.
If an $m\times p$-block of $A$ is mades of zeroes, then $m+p\le n$.

answered May 6 2011 at 7:29

Denis Serre
18.7k●1●31●93

And it's called Hall's marriage theorem. – darij grinberg May 6 2011 at 19:38

1

@darij, I'd rather say it can be proved using Hall's Marriage Theorem, or any of a number of other theorems equivalent to Hall's, including one, by Konig, that goes back at least 20 years before Hall, and was stated specifically in the context of matrices. – Gerry Myerson May 6 2011 at 22:55

Non-zero diagonal through permutation of rows

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.