Names for couple of structured matrices
$begingroup$
In my work I have come about two type of matrices:
1-Let $U_{np}in mathbb{R}^{ntimes p}$ be such that $(U_{np})_{ij}=p-j+1$, e.g. $U_{53}=begin{bmatrix}3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1
end{bmatrix}.$
2-Let $V_pinmathbb{R}^{ptimes p}$ be such that $(V_p)_{ij}=p-max(i,j)+1$, e.g. $V_5=begin{bmatrix}5 & 4 & 3 & 2 & 1\
4 & 4 & 3 & 2 & 1\
3 & 3 & 3 & 2 & 1\
2 & 2 & 2 & 2 & 1\
1 & 1 & 1 & 1 & 1
end{bmatrix}.$
I just want to know if these type of matrices have specific names or if anybody has seen them before, and could provide me with some references.
More context:
I am using the above matrices to simplify the following expressions:
$$begin{align*}
sum_{i=1}^{p}AI_{i;p} & =U_{np}odot A,text{ and }\
sum_{i=1}^{p}I_{i;p}A'AI_{i;p} & =V_{p}odot A'A,
end{align*},$$
where, $Ainmathbb{R}^{ntimes p}$, $odot$ is the is the Hadamard (element-wise) product, and $I_{i;p}$ is an $ptimes p$ matrix with all elements zero except the first $i$
diagonal terms or equivalently an $ptimes p$ identity matrix where
the last $p-i$ diagonal elements are zero. For example $I_{2;3}=begin{bmatrix}1 & 0 & 0\
0 & 1 & 0\
0 & 0 & 0
end{bmatrix}.$
linear-algebra matrices matrix-equations
asked Jan 6 at 4:45
RozaThRozaTh
324312
$endgroup$
add a comment |
$begingroup$
In my work I have come about two type of matrices:
1-Let $U_{np}in mathbb{R}^{ntimes p}$ be such that $(U_{np})_{ij}=p-j+1$, e.g. $U_{53}=begin{bmatrix}3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1
end{bmatrix}.$
2-Let $V_pinmathbb{R}^{ptimes p}$ be such that $(V_p)_{ij}=p-max(i,j)+1$, e.g. $V_5=begin{bmatrix}5 & 4 & 3 & 2 & 1\
4 & 4 & 3 & 2 & 1\
3 & 3 & 3 & 2 & 1\
2 & 2 & 2 & 2 & 1\
1 & 1 & 1 & 1 & 1
end{bmatrix}.$
I just want to know if these type of matrices have specific names or if anybody has seen them before, and could provide me with some references.
More context:
I am using the above matrices to simplify the following expressions:
$$begin{align*}
sum_{i=1}^{p}AI_{i;p} & =U_{np}odot A,text{ and }\
sum_{i=1}^{p}I_{i;p}A'AI_{i;p} & =V_{p}odot A'A,
end{align*},$$
where, $Ainmathbb{R}^{ntimes p}$, $odot$ is the is the Hadamard (element-wise) product, and $I_{i;p}$ is an $ptimes p$ matrix with all elements zero except the first $i$
diagonal terms or equivalently an $ptimes p$ identity matrix where
the last $p-i$ diagonal elements are zero. For example $I_{2;3}=begin{bmatrix}1 & 0 & 0\
0 & 1 & 0\
0 & 0 & 0
end{bmatrix}.$
linear-algebra matrices matrix-equations
asked Jan 6 at 4:45
RozaThRozaTh
324312
$endgroup$
add a comment |
$begingroup$
In my work I have come about two type of matrices:
1-Let $U_{np}in mathbb{R}^{ntimes p}$ be such that $(U_{np})_{ij}=p-j+1$, e.g. $U_{53}=begin{bmatrix}3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1
end{bmatrix}.$
2-Let $V_pinmathbb{R}^{ptimes p}$ be such that $(V_p)_{ij}=p-max(i,j)+1$, e.g. $V_5=begin{bmatrix}5 & 4 & 3 & 2 & 1\
4 & 4 & 3 & 2 & 1\
3 & 3 & 3 & 2 & 1\
2 & 2 & 2 & 2 & 1\
1 & 1 & 1 & 1 & 1
end{bmatrix}.$
I just want to know if these type of matrices have specific names or if anybody has seen them before, and could provide me with some references.
More context:
I am using the above matrices to simplify the following expressions:
$$begin{align*}
sum_{i=1}^{p}AI_{i;p} & =U_{np}odot A,text{ and }\
sum_{i=1}^{p}I_{i;p}A'AI_{i;p} & =V_{p}odot A'A,
end{align*},$$
where, $Ainmathbb{R}^{ntimes p}$, $odot$ is the is the Hadamard (element-wise) product, and $I_{i;p}$ is an $ptimes p$ matrix with all elements zero except the first $i$
diagonal terms or equivalently an $ptimes p$ identity matrix where
the last $p-i$ diagonal elements are zero. For example $I_{2;3}=begin{bmatrix}1 & 0 & 0\
0 & 1 & 0\
0 & 0 & 0
end{bmatrix}.$
linear-algebra matrices matrix-equations
asked Jan 6 at 4:45
RozaThRozaTh
324312
$endgroup$
In my work I have come about two type of matrices:
1-Let $U_{np}in mathbb{R}^{ntimes p}$ be such that $(U_{np})_{ij}=p-j+1$, e.g. $U_{53}=begin{bmatrix}3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1\
3 & 2 & 1
end{bmatrix}.$
2-Let $V_pinmathbb{R}^{ptimes p}$ be such that $(V_p)_{ij}=p-max(i,j)+1$, e.g. $V_5=begin{bmatrix}5 & 4 & 3 & 2 & 1\
4 & 4 & 3 & 2 & 1\
3 & 3 & 3 & 2 & 1\
2 & 2 & 2 & 2 & 1\
1 & 1 & 1 & 1 & 1
end{bmatrix}.$
I just want to know if these type of matrices have specific names or if anybody has seen them before, and could provide me with some references.
More context:
I am using the above matrices to simplify the following expressions:
$$begin{align*}
sum_{i=1}^{p}AI_{i;p} & =U_{np}odot A,text{ and }\
sum_{i=1}^{p}I_{i;p}A'AI_{i;p} & =V_{p}odot A'A,
end{align*},$$
where, $Ainmathbb{R}^{ntimes p}$, $odot$ is the is the Hadamard (element-wise) product, and $I_{i;p}$ is an $ptimes p$ matrix with all elements zero except the first $i$
diagonal terms or equivalently an $ptimes p$ identity matrix where
the last $p-i$ diagonal elements are zero. For example $I_{2;3}=begin{bmatrix}1 & 0 & 0\
0 & 1 & 0\
0 & 0 & 0
end{bmatrix}.$
linear-algebra matrices matrix-equations
linear-algebra matrices matrix-equations
asked Jan 6 at 4:45
RozaThRozaTh
324312
asked Jan 6 at 4:45
RozaThRozaTh
324312
asked Jan 6 at 4:45
RozaThRozaTh
324312
asked Jan 6 at 4:45
RozaThRozaTh
324312
asked Jan 6 at 4:45
RozaThRozaTh
324312
324312
add a comment |
add a comment |
1 Answer
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$begingroup$
The first matrix is a rank-one matrix product of the column vector with 5 ones and the row vector $(1 2 3)$. I don't see what can be said more...
The second matrix is characterized by its inverse. This inverse is (one form of) the (tridiagonal) discrete second differentiation matrix with entries $-1, 2, -1$ (with an exception at the beginning). See for example slide $11$ of {http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture4.pdf}
You mention a coupling between these matrices. I don't see any...
answered Jan 6 at 6:03
Jean MarieJean Marie
30.9k42155
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add a comment |
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$begingroup$
The first matrix is a rank-one matrix product of the column vector with 5 ones and the row vector $(1 2 3)$. I don't see what can be said more...
The second matrix is characterized by its inverse. This inverse is (one form of) the (tridiagonal) discrete second differentiation matrix with entries $-1, 2, -1$ (with an exception at the beginning). See for example slide $11$ of {http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture4.pdf}
You mention a coupling between these matrices. I don't see any...
answered Jan 6 at 6:03
Jean MarieJean Marie
30.9k42155
$endgroup$
add a comment |
$begingroup$
The first matrix is a rank-one matrix product of the column vector with 5 ones and the row vector $(1 2 3)$. I don't see what can be said more...
The second matrix is characterized by its inverse. This inverse is (one form of) the (tridiagonal) discrete second differentiation matrix with entries $-1, 2, -1$ (with an exception at the beginning). See for example slide $11$ of {http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture4.pdf}
You mention a coupling between these matrices. I don't see any...
answered Jan 6 at 6:03
Jean MarieJean Marie
30.9k42155
$endgroup$
add a comment |
$begingroup$
The first matrix is a rank-one matrix product of the column vector with 5 ones and the row vector $(1 2 3)$. I don't see what can be said more...
The second matrix is characterized by its inverse. This inverse is (one form of) the (tridiagonal) discrete second differentiation matrix with entries $-1, 2, -1$ (with an exception at the beginning). See for example slide $11$ of {http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture4.pdf}
You mention a coupling between these matrices. I don't see any...
answered Jan 6 at 6:03
Jean MarieJean Marie
30.9k42155
$endgroup$
The first matrix is a rank-one matrix product of the column vector with 5 ones and the row vector $(1 2 3)$. I don't see what can be said more...
The second matrix is characterized by its inverse. This inverse is (one form of) the (tridiagonal) discrete second differentiation matrix with entries $-1, 2, -1$ (with an exception at the beginning). See for example slide $11$ of {http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture4.pdf}
You mention a coupling between these matrices. I don't see any...
answered Jan 6 at 6:03
Jean MarieJean Marie
30.9k42155
edited Jan 6 at 6:28
answered Jan 6 at 6:03
Jean MarieJean Marie
30.9k42155
answered Jan 6 at 6:03
Jean MarieJean Marie
30.9k42155
answered Jan 6 at 6:03
Jean MarieJean Marie
30.9k42155
30.9k42155
add a comment |
add a comment |
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