Easiest way to express reversed identity matrix?
$begingroup$
Consider the "reverse identity matrix", let's denoted it by $bf I'$, by which I mean the $ntimes n$ matrix with zeros everywhere except on the diagonal from bottom left to the top right (i.e. $(delta'_{ij})$ where $delta'_{ij} = 1$ if $i+j = n + 1$ and $0$ otherwise). For example with $n=4$,
$$ mathbf{I'} = begin{pmatrix}
0&0&0&1\
0&0&1&0\
0&1&0&0\
1&0&0&0
end{pmatrix}.
$$
Question: Is there an easy way to express $bf I'$ in terms of commonly used matrices?
We usually try to express everything using $bf I$, the usual identity matrix, and $bf J$, the matrix consisting of all ones, and maybe the adjacency matrices of common graphs such as $K_n$. Also it goes without saying that easy operations such as transpose and so on are allowed.
linear-algebra matrices graph-theory
asked Dec 11 '18 at 23:52
Luke CollinsLuke Collins
739418
$endgroup$
add a comment |
$begingroup$
Consider the "reverse identity matrix", let's denoted it by $bf I'$, by which I mean the $ntimes n$ matrix with zeros everywhere except on the diagonal from bottom left to the top right (i.e. $(delta'_{ij})$ where $delta'_{ij} = 1$ if $i+j = n + 1$ and $0$ otherwise). For example with $n=4$,
$$ mathbf{I'} = begin{pmatrix}
0&0&0&1\
0&0&1&0\
0&1&0&0\
1&0&0&0
end{pmatrix}.
$$
Question: Is there an easy way to express $bf I'$ in terms of commonly used matrices?
We usually try to express everything using $bf I$, the usual identity matrix, and $bf J$, the matrix consisting of all ones, and maybe the adjacency matrices of common graphs such as $K_n$. Also it goes without saying that easy operations such as transpose and so on are allowed.
linear-algebra matrices graph-theory
asked Dec 11 '18 at 23:52
Luke CollinsLuke Collins
739418
$endgroup$
1
$begingroup$
If you allow adjacency matrices of common graphs then for $n=2k$, $bf I'$ is an adjacency matrix of a matching $(1,2k), (2,2k-1),dots, (k,k+1)$.
$endgroup$
– Alex Ravsky
Dec 12 '18 at 3:59
add a comment |
$begingroup$
Consider the "reverse identity matrix", let's denoted it by $bf I'$, by which I mean the $ntimes n$ matrix with zeros everywhere except on the diagonal from bottom left to the top right (i.e. $(delta'_{ij})$ where $delta'_{ij} = 1$ if $i+j = n + 1$ and $0$ otherwise). For example with $n=4$,
$$ mathbf{I'} = begin{pmatrix}
0&0&0&1\
0&0&1&0\
0&1&0&0\
1&0&0&0
end{pmatrix}.
$$
Question: Is there an easy way to express $bf I'$ in terms of commonly used matrices?
We usually try to express everything using $bf I$, the usual identity matrix, and $bf J$, the matrix consisting of all ones, and maybe the adjacency matrices of common graphs such as $K_n$. Also it goes without saying that easy operations such as transpose and so on are allowed.
linear-algebra matrices graph-theory
asked Dec 11 '18 at 23:52
Luke CollinsLuke Collins
739418
$endgroup$
Consider the "reverse identity matrix", let's denoted it by $bf I'$, by which I mean the $ntimes n$ matrix with zeros everywhere except on the diagonal from bottom left to the top right (i.e. $(delta'_{ij})$ where $delta'_{ij} = 1$ if $i+j = n + 1$ and $0$ otherwise). For example with $n=4$,
$$ mathbf{I'} = begin{pmatrix}
0&0&0&1\
0&0&1&0\
0&1&0&0\
1&0&0&0
end{pmatrix}.
$$
Question: Is there an easy way to express $bf I'$ in terms of commonly used matrices?
We usually try to express everything using $bf I$, the usual identity matrix, and $bf J$, the matrix consisting of all ones, and maybe the adjacency matrices of common graphs such as $K_n$. Also it goes without saying that easy operations such as transpose and so on are allowed.
linear-algebra matrices graph-theory
linear-algebra matrices graph-theory
asked Dec 11 '18 at 23:52
Luke CollinsLuke Collins
739418
asked Dec 11 '18 at 23:52
Luke CollinsLuke Collins
739418
asked Dec 11 '18 at 23:52
Luke CollinsLuke Collins
739418
asked Dec 11 '18 at 23:52
Luke CollinsLuke Collins
739418
asked Dec 11 '18 at 23:52
Luke CollinsLuke Collins
739418
739418
1
$begingroup$
If you allow adjacency matrices of common graphs then for $n=2k$, $bf I'$ is an adjacency matrix of a matching $(1,2k), (2,2k-1),dots, (k,k+1)$.
$endgroup$
– Alex Ravsky
Dec 12 '18 at 3:59
add a comment |
1
$begingroup$
If you allow adjacency matrices of common graphs then for $n=2k$, $bf I'$ is an adjacency matrix of a matching $(1,2k), (2,2k-1),dots, (k,k+1)$.
$endgroup$
– Alex Ravsky
Dec 12 '18 at 3:59
1
1
$begingroup$
If you allow adjacency matrices of common graphs then for $n=2k$, $bf I'$ is an adjacency matrix of a matching $(1,2k), (2,2k-1),dots, (k,k+1)$.
$endgroup$
– Alex Ravsky
Dec 12 '18 at 3:59
$begingroup$
If you allow adjacency matrices of common graphs then for $n=2k$, $bf I'$ is an adjacency matrix of a matching $(1,2k), (2,2k-1),dots, (k,k+1)$.
$endgroup$
– Alex Ravsky
Dec 12 '18 at 3:59
add a comment |
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$begingroup$
If you allow adjacency matrices of common graphs then for $n=2k$, $bf I'$ is an adjacency matrix of a matching $(1,2k), (2,2k-1),dots, (k,k+1)$.
$endgroup$
– Alex Ravsky
Dec 12 '18 at 3:59