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How to find log of “sum of two matrices�
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I want to find log (A + B ) where A and B are matrices.
The context is that I want to find the Von Neumann entropy which is given by:
$Entropy = - Trace [rho log (rho) ]$
where $rho$ is a matrix.
Now, my problem is computing the V. N. entropy analytically when $rho = rho_0 + deltarho$. For this I need to know $log (rho_0 + deltarho)$
Note that all $rho$s are matrices.
linear-algebra matrices logarithms quantum-mechanics entropy
asked Jul 30 at 14:53
Saurabh Shringarpure
151136
 |Â
show 2 more comments
up vote
0
down vote
favorite
I want to find log (A + B ) where A and B are matrices.
The context is that I want to find the Von Neumann entropy which is given by:
$Entropy = - Trace [rho log (rho) ]$
where $rho$ is a matrix.
Now, my problem is computing the V. N. entropy analytically when $rho = rho_0 + deltarho$. For this I need to know $log (rho_0 + deltarho)$
Note that all $rho$s are matrices.
linear-algebra matrices logarithms quantum-mechanics entropy
asked Jul 30 at 14:53
Saurabh Shringarpure
151136
Try the taylor series of $log$
– xbh
Jul 30 at 14:55
Can you please elaborate?
– Saurabh Shringarpure
Jul 30 at 14:57
1
Use the taylor series of ln(x+1) , now you have a way to compute it by setting ln(a+b) = ln(Q + 1) , Q = a + b - 1.
– mick
Jul 30 at 14:59
1
@SaurabhShringarpure I've seen that $log (boldsymbol A) = mathrm Log(boldsymbol I + boldsymbol A) = sum_1^infty (-1)^n-1 boldsymbol A^n/n $ whenever $|boldsymbol A| < 1$. This might be applicable, but stick to your text if the $log$ was defined before.
– xbh
Jul 30 at 15:04
So by doing that, I get: $log(rho_0 + deltarho)=log(rho_0 + deltarho-1+1)=1 + [rho_0 + deltarho-1]-frac[rho_0 + deltarho-1]^22+...$
– Saurabh Shringarpure
Jul 30 at 15:10
 |Â
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I want to find log (A + B ) where A and B are matrices.
The context is that I want to find the Von Neumann entropy which is given by:
$Entropy = - Trace [rho log (rho) ]$
where $rho$ is a matrix.
Now, my problem is computing the V. N. entropy analytically when $rho = rho_0 + deltarho$. For this I need to know $log (rho_0 + deltarho)$
Note that all $rho$s are matrices.
linear-algebra matrices logarithms quantum-mechanics entropy
asked Jul 30 at 14:53
Saurabh Shringarpure
151136
I want to find log (A + B ) where A and B are matrices.
The context is that I want to find the Von Neumann entropy which is given by:
$Entropy = - Trace [rho log (rho) ]$
where $rho$ is a matrix.
Now, my problem is computing the V. N. entropy analytically when $rho = rho_0 + deltarho$. For this I need to know $log (rho_0 + deltarho)$
Note that all $rho$s are matrices.
linear-algebra matrices logarithms quantum-mechanics entropy
asked Jul 30 at 14:53
Saurabh Shringarpure
151136
asked Jul 30 at 14:53
Saurabh Shringarpure
151136
asked Jul 30 at 14:53
Saurabh Shringarpure
151136
asked Jul 30 at 14:53
Saurabh Shringarpure
151136
151136
Try the taylor series of $log$
– xbh
Jul 30 at 14:55
Can you please elaborate?
– Saurabh Shringarpure
Jul 30 at 14:57
1
Use the taylor series of ln(x+1) , now you have a way to compute it by setting ln(a+b) = ln(Q + 1) , Q = a + b - 1.
– mick
Jul 30 at 14:59
1
@SaurabhShringarpure I've seen that $log (boldsymbol A) = mathrm Log(boldsymbol I + boldsymbol A) = sum_1^infty (-1)^n-1 boldsymbol A^n/n $ whenever $|boldsymbol A| < 1$. This might be applicable, but stick to your text if the $log$ was defined before.
– xbh
Jul 30 at 15:04
So by doing that, I get: $log(rho_0 + deltarho)=log(rho_0 + deltarho-1+1)=1 + [rho_0 + deltarho-1]-frac[rho_0 + deltarho-1]^22+...$
– Saurabh Shringarpure
Jul 30 at 15:10
 |Â
show 2 more comments
Try the taylor series of $log$
– xbh
Jul 30 at 14:55
Can you please elaborate?
– Saurabh Shringarpure
Jul 30 at 14:57
1
Use the taylor series of ln(x+1) , now you have a way to compute it by setting ln(a+b) = ln(Q + 1) , Q = a + b - 1.
– mick
Jul 30 at 14:59
1
@SaurabhShringarpure I've seen that $log (boldsymbol A) = mathrm Log(boldsymbol I + boldsymbol A) = sum_1^infty (-1)^n-1 boldsymbol A^n/n $ whenever $|boldsymbol A| < 1$. This might be applicable, but stick to your text if the $log$ was defined before.
– xbh
Jul 30 at 15:04
So by doing that, I get: $log(rho_0 + deltarho)=log(rho_0 + deltarho-1+1)=1 + [rho_0 + deltarho-1]-frac[rho_0 + deltarho-1]^22+...$
– Saurabh Shringarpure
Jul 30 at 15:10
Try the taylor series of $log$
– xbh
Jul 30 at 14:55
Try the taylor series of $log$
– xbh
Jul 30 at 14:55
Can you please elaborate?
– Saurabh Shringarpure
Jul 30 at 14:57
Can you please elaborate?
– Saurabh Shringarpure
Jul 30 at 14:57
1
1
Use the taylor series of ln(x+1) , now you have a way to compute it by setting ln(a+b) = ln(Q + 1) , Q = a + b - 1.
– mick
Jul 30 at 14:59
Use the taylor series of ln(x+1) , now you have a way to compute it by setting ln(a+b) = ln(Q + 1) , Q = a + b - 1.
– mick
Jul 30 at 14:59
1
1
@SaurabhShringarpure I've seen that $log (boldsymbol A) = mathrm Log(boldsymbol I + boldsymbol A) = sum_1^infty (-1)^n-1 boldsymbol A^n/n $ whenever $|boldsymbol A| < 1$. This might be applicable, but stick to your text if the $log$ was defined before.
– xbh
Jul 30 at 15:04
@SaurabhShringarpure I've seen that $log (boldsymbol A) = mathrm Log(boldsymbol I + boldsymbol A) = sum_1^infty (-1)^n-1 boldsymbol A^n/n $ whenever $|boldsymbol A| < 1$. This might be applicable, but stick to your text if the $log$ was defined before.
– xbh
Jul 30 at 15:04
So by doing that, I get: $log(rho_0 + deltarho)=log(rho_0 + deltarho-1+1)=1 + [rho_0 + deltarho-1]-frac[rho_0 + deltarho-1]^22+...$
– Saurabh Shringarpure
Jul 30 at 15:10
So by doing that, I get: $log(rho_0 + deltarho)=log(rho_0 + deltarho-1+1)=1 + [rho_0 + deltarho-1]-frac[rho_0 + deltarho-1]^22+...$
– Saurabh Shringarpure
Jul 30 at 15:10
 |Â
show 2 more comments
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Try the taylor series of $log$
– xbh
Jul 30 at 14:55
Can you please elaborate?
– Saurabh Shringarpure
Jul 30 at 14:57
1
Use the taylor series of ln(x+1) , now you have a way to compute it by setting ln(a+b) = ln(Q + 1) , Q = a + b - 1.
– mick
Jul 30 at 14:59
1
@SaurabhShringarpure I've seen that $log (boldsymbol A) = mathrm Log(boldsymbol I + boldsymbol A) = sum_1^infty (-1)^n-1 boldsymbol A^n/n $ whenever $|boldsymbol A| < 1$. This might be applicable, but stick to your text if the $log$ was defined before.
– xbh
Jul 30 at 15:04
So by doing that, I get: $log(rho_0 + deltarho)=log(rho_0 + deltarho-1+1)=1 + [rho_0 + deltarho-1]-frac[rho_0 + deltarho-1]^22+...$
– Saurabh Shringarpure
Jul 30 at 15:10