Does this form of matrix have a name?
$begingroup$
I'm looking for the name of this kind of $n$-by-$n$ matrix:
$$left(begin{array}{cccc}
-s_1 & b_{12} & b_{13} & b_{14} \
b_{21} & -s_2 & b_{23} & b_{24} \
b_{31} & b_{32} & -s_3 & b_{34} \
b_{41} & b_{42} & b_{43} & -s_4
end{array}right)$$
where $s_i = sum_{jneq i} b_{ij}$. This is basically so I can google it, so by name I really mean an appropriate "search string".
linear-algebra matrices
asked Jun 12 '13 at 11:25
LucasLucas
1,032720
$endgroup$
add a comment |
$begingroup$
I'm looking for the name of this kind of $n$-by-$n$ matrix:
$$left(begin{array}{cccc}
-s_1 & b_{12} & b_{13} & b_{14} \
b_{21} & -s_2 & b_{23} & b_{24} \
b_{31} & b_{32} & -s_3 & b_{34} \
b_{41} & b_{42} & b_{43} & -s_4
end{array}right)$$
where $s_i = sum_{jneq i} b_{ij}$. This is basically so I can google it, so by name I really mean an appropriate "search string".
linear-algebra matrices
asked Jun 12 '13 at 11:25
LucasLucas
1,032720
$endgroup$
$begingroup$
I think it's the Q-matrix of a continuous time Markov chain, but I'm not sure...
$endgroup$
– user71815
Jun 12 '13 at 11:29
$begingroup$
Sounds about right
$endgroup$
– Lucas
Jun 12 '13 at 15:50
$begingroup$
It reminds of a adjacency matrix (see wiki). Your matrix is minus the neighborhood matrix for the Besag model (slide 10).
$endgroup$
– Rafael
Dec 14 '18 at 22:58
add a comment |
$begingroup$
I'm looking for the name of this kind of $n$-by-$n$ matrix:
$$left(begin{array}{cccc}
-s_1 & b_{12} & b_{13} & b_{14} \
b_{21} & -s_2 & b_{23} & b_{24} \
b_{31} & b_{32} & -s_3 & b_{34} \
b_{41} & b_{42} & b_{43} & -s_4
end{array}right)$$
where $s_i = sum_{jneq i} b_{ij}$. This is basically so I can google it, so by name I really mean an appropriate "search string".
linear-algebra matrices
asked Jun 12 '13 at 11:25
LucasLucas
1,032720
$endgroup$
I'm looking for the name of this kind of $n$-by-$n$ matrix:
$$left(begin{array}{cccc}
-s_1 & b_{12} & b_{13} & b_{14} \
b_{21} & -s_2 & b_{23} & b_{24} \
b_{31} & b_{32} & -s_3 & b_{34} \
b_{41} & b_{42} & b_{43} & -s_4
end{array}right)$$
where $s_i = sum_{jneq i} b_{ij}$. This is basically so I can google it, so by name I really mean an appropriate "search string".
linear-algebra matrices
linear-algebra matrices
asked Jun 12 '13 at 11:25
LucasLucas
1,032720
asked Jun 12 '13 at 11:25
LucasLucas
1,032720
asked Jun 12 '13 at 11:25
LucasLucas
1,032720
asked Jun 12 '13 at 11:25
LucasLucas
1,032720
asked Jun 12 '13 at 11:25
LucasLucas
1,032720
1,032720
$begingroup$
I think it's the Q-matrix of a continuous time Markov chain, but I'm not sure...
$endgroup$
– user71815
Jun 12 '13 at 11:29
$begingroup$
Sounds about right
$endgroup$
– Lucas
Jun 12 '13 at 15:50
$begingroup$
It reminds of a adjacency matrix (see wiki). Your matrix is minus the neighborhood matrix for the Besag model (slide 10).
$endgroup$
– Rafael
Dec 14 '18 at 22:58
add a comment |
$begingroup$
I think it's the Q-matrix of a continuous time Markov chain, but I'm not sure...
$endgroup$
– user71815
Jun 12 '13 at 11:29
$begingroup$
Sounds about right
$endgroup$
– Lucas
Jun 12 '13 at 15:50
$begingroup$
It reminds of a adjacency matrix (see wiki). Your matrix is minus the neighborhood matrix for the Besag model (slide 10).
$endgroup$
– Rafael
Dec 14 '18 at 22:58
$begingroup$
I think it's the Q-matrix of a continuous time Markov chain, but I'm not sure...
$endgroup$
– user71815
Jun 12 '13 at 11:29
$begingroup$
I think it's the Q-matrix of a continuous time Markov chain, but I'm not sure...
$endgroup$
– user71815
Jun 12 '13 at 11:29
$begingroup$
Sounds about right
$endgroup$
– Lucas
Jun 12 '13 at 15:50
$begingroup$
Sounds about right
$endgroup$
– Lucas
Jun 12 '13 at 15:50
$begingroup$
It reminds of a adjacency matrix (see wiki). Your matrix is minus the neighborhood matrix for the Besag model (slide 10).
$endgroup$
– Rafael
Dec 14 '18 at 22:58
$begingroup$
It reminds of a adjacency matrix (see wiki). Your matrix is minus the neighborhood matrix for the Besag model (slide 10).
$endgroup$
– Rafael
Dec 14 '18 at 22:58
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
We have diagonally dominant matrices if absolute values of diagonal elements are larger than sum of absolute values of the others in that row.
Such matrices are invertible, ie. nonzero determinant.
answered May 4 '16 at 6:36
Behnam EsmayliBehnam Esmayli
1,956515
$endgroup$
add a comment |
$begingroup$
In your matrix the row sums are null.
In a Stochastic Matrix they are equal to $1$, with the elements being non-negative.
If your $b_{i,j}$'s are also non negative, then your matrix is essentially a diagonal matrix $times$ (a stochastic matrix minus $I$).
And that in brackets is a matrix that , in a Markov chain, gives the difference between the configuration at step $n+1$ wrt that at step $n$.
answered Dec 14 '18 at 23:56
G CabG Cab
19.4k31238
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
We have diagonally dominant matrices if absolute values of diagonal elements are larger than sum of absolute values of the others in that row.
Such matrices are invertible, ie. nonzero determinant.
answered May 4 '16 at 6:36
Behnam EsmayliBehnam Esmayli
1,956515
$endgroup$
add a comment |
$begingroup$
We have diagonally dominant matrices if absolute values of diagonal elements are larger than sum of absolute values of the others in that row.
Such matrices are invertible, ie. nonzero determinant.
answered May 4 '16 at 6:36
Behnam EsmayliBehnam Esmayli
1,956515
$endgroup$
add a comment |
$begingroup$
We have diagonally dominant matrices if absolute values of diagonal elements are larger than sum of absolute values of the others in that row.
Such matrices are invertible, ie. nonzero determinant.
answered May 4 '16 at 6:36
Behnam EsmayliBehnam Esmayli
1,956515
$endgroup$
We have diagonally dominant matrices if absolute values of diagonal elements are larger than sum of absolute values of the others in that row.
Such matrices are invertible, ie. nonzero determinant.
answered May 4 '16 at 6:36
Behnam EsmayliBehnam Esmayli
1,956515
answered May 4 '16 at 6:36
Behnam EsmayliBehnam Esmayli
1,956515
answered May 4 '16 at 6:36
Behnam EsmayliBehnam Esmayli
1,956515
answered May 4 '16 at 6:36
Behnam EsmayliBehnam Esmayli
1,956515
1,956515
add a comment |
add a comment |
$begingroup$
In your matrix the row sums are null.
In a Stochastic Matrix they are equal to $1$, with the elements being non-negative.
If your $b_{i,j}$'s are also non negative, then your matrix is essentially a diagonal matrix $times$ (a stochastic matrix minus $I$).
And that in brackets is a matrix that , in a Markov chain, gives the difference between the configuration at step $n+1$ wrt that at step $n$.
answered Dec 14 '18 at 23:56
G CabG Cab
19.4k31238
$endgroup$
add a comment |
$begingroup$
In your matrix the row sums are null.
In a Stochastic Matrix they are equal to $1$, with the elements being non-negative.
If your $b_{i,j}$'s are also non negative, then your matrix is essentially a diagonal matrix $times$ (a stochastic matrix minus $I$).
And that in brackets is a matrix that , in a Markov chain, gives the difference between the configuration at step $n+1$ wrt that at step $n$.
answered Dec 14 '18 at 23:56
G CabG Cab
19.4k31238
$endgroup$
add a comment |
$begingroup$
In your matrix the row sums are null.
In a Stochastic Matrix they are equal to $1$, with the elements being non-negative.
If your $b_{i,j}$'s are also non negative, then your matrix is essentially a diagonal matrix $times$ (a stochastic matrix minus $I$).
And that in brackets is a matrix that , in a Markov chain, gives the difference between the configuration at step $n+1$ wrt that at step $n$.
answered Dec 14 '18 at 23:56
G CabG Cab
19.4k31238
$endgroup$
In your matrix the row sums are null.
In a Stochastic Matrix they are equal to $1$, with the elements being non-negative.
If your $b_{i,j}$'s are also non negative, then your matrix is essentially a diagonal matrix $times$ (a stochastic matrix minus $I$).
And that in brackets is a matrix that , in a Markov chain, gives the difference between the configuration at step $n+1$ wrt that at step $n$.
answered Dec 14 '18 at 23:56
G CabG Cab
19.4k31238
answered Dec 14 '18 at 23:56
G CabG Cab
19.4k31238
answered Dec 14 '18 at 23:56
G CabG Cab
19.4k31238
answered Dec 14 '18 at 23:56
G CabG Cab
19.4k31238
19.4k31238
add a comment |
add a comment |
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$begingroup$
I think it's the Q-matrix of a continuous time Markov chain, but I'm not sure...
$endgroup$
– user71815
Jun 12 '13 at 11:29
$begingroup$
Sounds about right
$endgroup$
– Lucas
Jun 12 '13 at 15:50
$begingroup$
It reminds of a adjacency matrix (see wiki). Your matrix is minus the neighborhood matrix for the Besag model (slide 10).
$endgroup$
– Rafael
Dec 14 '18 at 22:58