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 $\uparrow a=\{p\in P | a\leq p \}$. ...
In what follows, I transcribe an example that illustrates the application of the Nehari Manifold method to solve a nonlinear partial differential equation contained in the text (in Portuguese) that I am reading. My question is: Why can we conclude that $\Phi(u) \geq 0$ in the proof of Lemma 1.2 ...
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...
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