Exponential of a symmetric, block tridiagonal matrix (with zero on the diagonal) The Next CEO...
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Exponential of a symmetric, block tridiagonal matrix (with zero on the diagonal)
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Is there an analytic result for the exponential of a symmetric, tridiagonal matrix (the diagonal can be zero, if this helps). Moreover, if it simplifies the result, the $alpha$'s matrices can be (asymmetric) banded.
Here is an example of such a matrix:
$$begin{bmatrix} 0 & alpha_2 & & & \
alpha_2 & 0 & alpha_3 & & & \
& & ddots & & \
& & alpha_{n-1} & 0 & alpha_n \
& & & alpha_n & 0
end{bmatrix}$$
I am aware that there are numerical means to do this (more or less accurate), e.g. most notably the software package EXPOKIT.
matrices exponentiation
asked Jun 6 '16 at 7:34
ChipChip
99939
$endgroup$
add a comment |
$begingroup$
Is there an analytic result for the exponential of a symmetric, tridiagonal matrix (the diagonal can be zero, if this helps). Moreover, if it simplifies the result, the $alpha$'s matrices can be (asymmetric) banded.
Here is an example of such a matrix:
$$begin{bmatrix} 0 & alpha_2 & & & \
alpha_2 & 0 & alpha_3 & & & \
& & ddots & & \
& & alpha_{n-1} & 0 & alpha_n \
& & & alpha_n & 0
end{bmatrix}$$
I am aware that there are numerical means to do this (more or less accurate), e.g. most notably the software package EXPOKIT.
matrices exponentiation
asked Jun 6 '16 at 7:34
ChipChip
99939
$endgroup$
$begingroup$
What does 'purely complex' mean? The above appears to be purely imaginary.
$endgroup$
– Semiclassical
Jun 6 '16 at 16:32
$begingroup$
I'm not sure I understand your notation, but what's wrong with $(I cosh alpha_2+sigma_1sinhalpha_2)oplus ... oplus (I cosh alpha_n+sigma_1sinhalpha_n)$, where $sigma_1 = begin{pmatrix} 0&1\ 1&0 end{pmatrix}$ is the usual Pauli matrix?
$endgroup$
– Cosmas Zachos
Jan 3 '18 at 19:41
add a comment |
$begingroup$
Is there an analytic result for the exponential of a symmetric, tridiagonal matrix (the diagonal can be zero, if this helps). Moreover, if it simplifies the result, the $alpha$'s matrices can be (asymmetric) banded.
Here is an example of such a matrix:
$$begin{bmatrix} 0 & alpha_2 & & & \
alpha_2 & 0 & alpha_3 & & & \
& & ddots & & \
& & alpha_{n-1} & 0 & alpha_n \
& & & alpha_n & 0
end{bmatrix}$$
I am aware that there are numerical means to do this (more or less accurate), e.g. most notably the software package EXPOKIT.
matrices exponentiation
asked Jun 6 '16 at 7:34
ChipChip
99939
$endgroup$
Is there an analytic result for the exponential of a symmetric, tridiagonal matrix (the diagonal can be zero, if this helps). Moreover, if it simplifies the result, the $alpha$'s matrices can be (asymmetric) banded.
Here is an example of such a matrix:
$$begin{bmatrix} 0 & alpha_2 & & & \
alpha_2 & 0 & alpha_3 & & & \
& & ddots & & \
& & alpha_{n-1} & 0 & alpha_n \
& & & alpha_n & 0
end{bmatrix}$$
I am aware that there are numerical means to do this (more or less accurate), e.g. most notably the software package EXPOKIT.
matrices exponentiation
matrices exponentiation
asked Jun 6 '16 at 7:34
ChipChip
99939
asked Jun 6 '16 at 7:34
ChipChip
99939
edited Mar 17 at 5:32
Chip
asked Jun 6 '16 at 7:34
ChipChip
99939
asked Jun 6 '16 at 7:34
ChipChip
99939
asked Jun 6 '16 at 7:34
ChipChip
99939
99939
$begingroup$
What does 'purely complex' mean? The above appears to be purely imaginary.
$endgroup$
– Semiclassical
Jun 6 '16 at 16:32
$begingroup$
I'm not sure I understand your notation, but what's wrong with $(I cosh alpha_2+sigma_1sinhalpha_2)oplus ... oplus (I cosh alpha_n+sigma_1sinhalpha_n)$, where $sigma_1 = begin{pmatrix} 0&1\ 1&0 end{pmatrix}$ is the usual Pauli matrix?
$endgroup$
– Cosmas Zachos
Jan 3 '18 at 19:41
add a comment |
$begingroup$
What does 'purely complex' mean? The above appears to be purely imaginary.
$endgroup$
– Semiclassical
Jun 6 '16 at 16:32
$begingroup$
I'm not sure I understand your notation, but what's wrong with $(I cosh alpha_2+sigma_1sinhalpha_2)oplus ... oplus (I cosh alpha_n+sigma_1sinhalpha_n)$, where $sigma_1 = begin{pmatrix} 0&1\ 1&0 end{pmatrix}$ is the usual Pauli matrix?
$endgroup$
– Cosmas Zachos
Jan 3 '18 at 19:41
$begingroup$
What does 'purely complex' mean? The above appears to be purely imaginary.
$endgroup$
– Semiclassical
Jun 6 '16 at 16:32
$begingroup$
What does 'purely complex' mean? The above appears to be purely imaginary.
$endgroup$
– Semiclassical
Jun 6 '16 at 16:32
$begingroup$
I'm not sure I understand your notation, but what's wrong with $(I cosh alpha_2+sigma_1sinhalpha_2)oplus ... oplus (I cosh alpha_n+sigma_1sinhalpha_n)$, where $sigma_1 = begin{pmatrix} 0&1\ 1&0 end{pmatrix}$ is the usual Pauli matrix?
$endgroup$
– Cosmas Zachos
Jan 3 '18 at 19:41
$begingroup$
I'm not sure I understand your notation, but what's wrong with $(I cosh alpha_2+sigma_1sinhalpha_2)oplus ... oplus (I cosh alpha_n+sigma_1sinhalpha_n)$, where $sigma_1 = begin{pmatrix} 0&1\ 1&0 end{pmatrix}$ is the usual Pauli matrix?
$endgroup$
– Cosmas Zachos
Jan 3 '18 at 19:41
add a comment |
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$begingroup$
What does 'purely complex' mean? The above appears to be purely imaginary.
$endgroup$
– Semiclassical
Jun 6 '16 at 16:32
$begingroup$
I'm not sure I understand your notation, but what's wrong with $(I cosh alpha_2+sigma_1sinhalpha_2)oplus ... oplus (I cosh alpha_n+sigma_1sinhalpha_n)$, where $sigma_1 = begin{pmatrix} 0&1\ 1&0 end{pmatrix}$ is the usual Pauli matrix?
$endgroup$
– Cosmas Zachos
Jan 3 '18 at 19:41