We know that the product of two diagonal matrices forms another diagonal matrix, since we just multiply the entries.
So my question is, does the converse necessarily hold? In other words, if I have a diagonal matrix, did it necessarily come from the product of two other diagonal matrices?
My gut feeling tells me "No" since in Linear Algebra, all sorts of 'intuition' seem to go wrong.
But I would like a confirmation and perhaps a counter example would be nice.