Multiplying band matrices
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Consider band matrices of the form
$$M_n:=
begin{pmatrix}
1 & 0 & 0 & 0 & 0 \
n & 2 & 0 & 0 & 0 \
0 & n-1 & 3 & 0 & \
0 & 0 & ddots & ddots & 0\
0 & 0 & 0 & 2 & n \
0 & 0 & 0 & 0 & 1 \
end{pmatrix}$$
I realized using CAS that if I multiplify $M_{n+a}cdots M_{n+1}M_n$ I will eventually get a matrix where the upper half has decreasing rows (when reading from left to right).
For example, for $n=4$ we have
$$M_7cdot M_6cdot M_5cdot M_4=
begin{pmatrix}
1 & 0 & 0 & 0 \
71 & 16 & 0 & 0 \
531 & 261 & 81 & 0 \
821 & 771 & 551 & 256 \
256 & 551 & 771 & 821 \
0 & 81 & 261 & 531 \
0 & 0 & 16 & 71 \
0 & 0 & 0 & 1 \
end{pmatrix}$$
and $821 > 771 > 551 > 256$. How can I prove such property?
For those who are curious, the property appears to start working after $a=(n-2)^2-1$ steps according to the numerical computations for low values of $n$.
matrices
asked Nov 17 at 23:36
user61170
12
add a comment |
up vote
0
down vote
favorite
Consider band matrices of the form
$$M_n:=
begin{pmatrix}
1 & 0 & 0 & 0 & 0 \
n & 2 & 0 & 0 & 0 \
0 & n-1 & 3 & 0 & \
0 & 0 & ddots & ddots & 0\
0 & 0 & 0 & 2 & n \
0 & 0 & 0 & 0 & 1 \
end{pmatrix}$$
I realized using CAS that if I multiplify $M_{n+a}cdots M_{n+1}M_n$ I will eventually get a matrix where the upper half has decreasing rows (when reading from left to right).
For example, for $n=4$ we have
$$M_7cdot M_6cdot M_5cdot M_4=
begin{pmatrix}
1 & 0 & 0 & 0 \
71 & 16 & 0 & 0 \
531 & 261 & 81 & 0 \
821 & 771 & 551 & 256 \
256 & 551 & 771 & 821 \
0 & 81 & 261 & 531 \
0 & 0 & 16 & 71 \
0 & 0 & 0 & 1 \
end{pmatrix}$$
and $821 > 771 > 551 > 256$. How can I prove such property?
For those who are curious, the property appears to start working after $a=(n-2)^2-1$ steps according to the numerical computations for low values of $n$.
matrices
asked Nov 17 at 23:36
user61170
12
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider band matrices of the form
$$M_n:=
begin{pmatrix}
1 & 0 & 0 & 0 & 0 \
n & 2 & 0 & 0 & 0 \
0 & n-1 & 3 & 0 & \
0 & 0 & ddots & ddots & 0\
0 & 0 & 0 & 2 & n \
0 & 0 & 0 & 0 & 1 \
end{pmatrix}$$
I realized using CAS that if I multiplify $M_{n+a}cdots M_{n+1}M_n$ I will eventually get a matrix where the upper half has decreasing rows (when reading from left to right).
For example, for $n=4$ we have
$$M_7cdot M_6cdot M_5cdot M_4=
begin{pmatrix}
1 & 0 & 0 & 0 \
71 & 16 & 0 & 0 \
531 & 261 & 81 & 0 \
821 & 771 & 551 & 256 \
256 & 551 & 771 & 821 \
0 & 81 & 261 & 531 \
0 & 0 & 16 & 71 \
0 & 0 & 0 & 1 \
end{pmatrix}$$
and $821 > 771 > 551 > 256$. How can I prove such property?
For those who are curious, the property appears to start working after $a=(n-2)^2-1$ steps according to the numerical computations for low values of $n$.
matrices
asked Nov 17 at 23:36
user61170
12
Consider band matrices of the form
$$M_n:=
begin{pmatrix}
1 & 0 & 0 & 0 & 0 \
n & 2 & 0 & 0 & 0 \
0 & n-1 & 3 & 0 & \
0 & 0 & ddots & ddots & 0\
0 & 0 & 0 & 2 & n \
0 & 0 & 0 & 0 & 1 \
end{pmatrix}$$
I realized using CAS that if I multiplify $M_{n+a}cdots M_{n+1}M_n$ I will eventually get a matrix where the upper half has decreasing rows (when reading from left to right).
For example, for $n=4$ we have
$$M_7cdot M_6cdot M_5cdot M_4=
begin{pmatrix}
1 & 0 & 0 & 0 \
71 & 16 & 0 & 0 \
531 & 261 & 81 & 0 \
821 & 771 & 551 & 256 \
256 & 551 & 771 & 821 \
0 & 81 & 261 & 531 \
0 & 0 & 16 & 71 \
0 & 0 & 0 & 1 \
end{pmatrix}$$
and $821 > 771 > 551 > 256$. How can I prove such property?
For those who are curious, the property appears to start working after $a=(n-2)^2-1$ steps according to the numerical computations for low values of $n$.
matrices
matrices
asked Nov 17 at 23:36
user61170
12
asked Nov 17 at 23:36
user61170
12
asked Nov 17 at 23:36
user61170
12
asked Nov 17 at 23:36
user61170
12
asked Nov 17 at 23:36
user61170
12
12
add a comment |
add a comment |
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