I am working through an applied category text. In it he makes several definitions: Definition 1: Let $(P, \leq)$ be a preorder. Then an upper set for P is a set such that if $p\in P$ and $p\leq q$, then $q\in P$. Definition 2: For a preorder as above, let $\uparrow a=\{p\in P | a\leq p \}$. ...

When we want to define transformations using permutations, what are the subtle differencies betwen the use of permutation matrices, and the use of permutations? Say I want to define a way to shuffle a sequence of numbers. Shoud I define my transformation with permutation matrices, or only with t...

The set $P$ of $n \times n$ permutation matrices spans a subspace of dimension $(n-1)^2+1$ within, say, the $n \times n$ complex matrices. Is there another description of this space? In particular, I am interested in a description of a subset of the permutation matrices which will form a basis....

Just simple question: Can anyone provide a list of types of permutation matrices that commute (with the matrices of the same type)? for one, I can think of rotation matrix... (Oh, wait. it isn't really permutation matrix..)

$\newcommand\mat{\mathbf}$A permutation matrix is a matrix whose columns are a permutation of the columns of the identity matrix $\mat I$. In other words, a permutation matrix is a matrix $\mat P$ with precisely one $1$ per row/column and zeros everywhere else. A few easy observations about per...

Suppose I have a $n\times n$ matrix $A$. Can I, by using only pre- and post-multiplication by permutation matrices, permute all the elements of $A$? That is, there should be no binding conditions, like $a_{11}$ will always be to the left of $a_{n1}$, etc. This seems to be intuitively obvious. Wh...