Mixing
Why is $A$ on the left hand side yet $B$ is on the right hand side when we evaluate $ fracddt(e^tACe^tB) = Ae^tACe^tB + e^tACe^tBB$?
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
I came across the following expression where $A:Yto Y$, $B:Xto X$, and $C:Xto Y$ with $X$ and $Y$ being Banach spaces:
$$
fracddt(e^tACe^tB) = Ae^tACe^tB + e^tACe^tBB.
$$
So we have the product rule followed by the chain rule. Regarding the expression on the right hand side, why is $A$ on the left hand side in the first expression, yet $B$ is on the right hand side in the second expression?
functional-analysis operator-theory exponential-function banach-spaces
asked Jul 15 at 9:50
csss
1,22811221
add a comment |Â
up vote
0
down vote
favorite
I came across the following expression where $A:Yto Y$, $B:Xto X$, and $C:Xto Y$ with $X$ and $Y$ being Banach spaces:
$$
fracddt(e^tACe^tB) = Ae^tACe^tB + e^tACe^tBB.
$$
So we have the product rule followed by the chain rule. Regarding the expression on the right hand side, why is $A$ on the left hand side in the first expression, yet $B$ is on the right hand side in the second expression?
functional-analysis operator-theory exponential-function banach-spaces
asked Jul 15 at 9:50
csss
1,22811221
1
$B$ is on the right, for the same reason as $A$ is on the left.
– Lord Shark the Unknown
Jul 15 at 9:54
Strictly speaking, you might write the second term as $ mathrm e^tACBmathrm e^tB$, but I guess it's for symmetry reasons. Don't forget $B$ and $mathrm e^tB$ commute.
– Bernard
Jul 15 at 9:55
Why do they commute?
– csss
Jul 31 at 15:06
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I came across the following expression where $A:Yto Y$, $B:Xto X$, and $C:Xto Y$ with $X$ and $Y$ being Banach spaces:
$$
fracddt(e^tACe^tB) = Ae^tACe^tB + e^tACe^tBB.
$$
So we have the product rule followed by the chain rule. Regarding the expression on the right hand side, why is $A$ on the left hand side in the first expression, yet $B$ is on the right hand side in the second expression?
functional-analysis operator-theory exponential-function banach-spaces
asked Jul 15 at 9:50
csss
1,22811221
I came across the following expression where $A:Yto Y$, $B:Xto X$, and $C:Xto Y$ with $X$ and $Y$ being Banach spaces:
$$
fracddt(e^tACe^tB) = Ae^tACe^tB + e^tACe^tBB.
$$
So we have the product rule followed by the chain rule. Regarding the expression on the right hand side, why is $A$ on the left hand side in the first expression, yet $B$ is on the right hand side in the second expression?
functional-analysis operator-theory exponential-function banach-spaces
asked Jul 15 at 9:50
csss
1,22811221
asked Jul 15 at 9:50
csss
1,22811221
asked Jul 15 at 9:50
csss
1,22811221
asked Jul 15 at 9:50
csss
1,22811221
1,22811221
1
$B$ is on the right, for the same reason as $A$ is on the left.
– Lord Shark the Unknown
Jul 15 at 9:54
Strictly speaking, you might write the second term as $ mathrm e^tACBmathrm e^tB$, but I guess it's for symmetry reasons. Don't forget $B$ and $mathrm e^tB$ commute.
– Bernard
Jul 15 at 9:55
Why do they commute?
– csss
Jul 31 at 15:06
add a comment |Â
1
$B$ is on the right, for the same reason as $A$ is on the left.
– Lord Shark the Unknown
Jul 15 at 9:54
Strictly speaking, you might write the second term as $ mathrm e^tACBmathrm e^tB$, but I guess it's for symmetry reasons. Don't forget $B$ and $mathrm e^tB$ commute.
– Bernard
Jul 15 at 9:55
Why do they commute?
– csss
Jul 31 at 15:06
1
1
$B$ is on the right, for the same reason as $A$ is on the left.
– Lord Shark the Unknown
Jul 15 at 9:54
$B$ is on the right, for the same reason as $A$ is on the left.
– Lord Shark the Unknown
Jul 15 at 9:54
Strictly speaking, you might write the second term as $ mathrm e^tACBmathrm e^tB$, but I guess it's for symmetry reasons. Don't forget $B$ and $mathrm e^tB$ commute.
– Bernard
Jul 15 at 9:55
Strictly speaking, you might write the second term as $ mathrm e^tACBmathrm e^tB$, but I guess it's for symmetry reasons. Don't forget $B$ and $mathrm e^tB$ commute.
– Bernard
Jul 15 at 9:55
Why do they commute?
– csss
Jul 31 at 15:06
Why do they commute?
– csss
Jul 31 at 15:06
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
For no special reason. $e^A$ and $A$ commute for any matrix $A$.
In any case, $B$ should be to the right of $C$.
answered Jul 15 at 9:55
Julián Aguirre
64.7k23894
Why do they commute?
– csss
Jul 31 at 15:06
$$A,e^A=Asum_n=0^inftyfracA^nn!=sum_n=0^inftyfracA^n+1n!=e^A,A.$$
– Julián Aguirre
Jul 31 at 15:08
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
For no special reason. $e^A$ and $A$ commute for any matrix $A$.
In any case, $B$ should be to the right of $C$.
answered Jul 15 at 9:55
Julián Aguirre
64.7k23894
Why do they commute?
– csss
Jul 31 at 15:06
$$A,e^A=Asum_n=0^inftyfracA^nn!=sum_n=0^inftyfracA^n+1n!=e^A,A.$$
– Julián Aguirre
Jul 31 at 15:08
add a comment |Â
up vote
1
down vote
accepted
For no special reason. $e^A$ and $A$ commute for any matrix $A$.
In any case, $B$ should be to the right of $C$.
answered Jul 15 at 9:55
Julián Aguirre
64.7k23894
Why do they commute?
– csss
Jul 31 at 15:06
$$A,e^A=Asum_n=0^inftyfracA^nn!=sum_n=0^inftyfracA^n+1n!=e^A,A.$$
– Julián Aguirre
Jul 31 at 15:08
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
For no special reason. $e^A$ and $A$ commute for any matrix $A$.
In any case, $B$ should be to the right of $C$.
answered Jul 15 at 9:55
Julián Aguirre
64.7k23894
For no special reason. $e^A$ and $A$ commute for any matrix $A$.
In any case, $B$ should be to the right of $C$.
answered Jul 15 at 9:55
Julián Aguirre
64.7k23894
answered Jul 15 at 9:55
Julián Aguirre
64.7k23894
answered Jul 15 at 9:55
Julián Aguirre
64.7k23894
answered Jul 15 at 9:55
Julián Aguirre
64.7k23894
64.7k23894
Why do they commute?
– csss
Jul 31 at 15:06
$$A,e^A=Asum_n=0^inftyfracA^nn!=sum_n=0^inftyfracA^n+1n!=e^A,A.$$
– Julián Aguirre
Jul 31 at 15:08
add a comment |Â
Why do they commute?
– csss
Jul 31 at 15:06
$$A,e^A=Asum_n=0^inftyfracA^nn!=sum_n=0^inftyfracA^n+1n!=e^A,A.$$
– Julián Aguirre
Jul 31 at 15:08
Why do they commute?
– csss
Jul 31 at 15:06
Why do they commute?
– csss
Jul 31 at 15:06
$$A,e^A=Asum_n=0^inftyfracA^nn!=sum_n=0^inftyfracA^n+1n!=e^A,A.$$
– Julián Aguirre
Jul 31 at 15:08
$$A,e^A=Asum_n=0^inftyfracA^nn!=sum_n=0^inftyfracA^n+1n!=e^A,A.$$
– Julián Aguirre
Jul 31 at 15:08
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2852341%2fwhy-is-a-on-the-left-hand-side-yet-b-is-on-the-right-hand-side-when-we-evalu%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
$B$ is on the right, for the same reason as $A$ is on the left.
– Lord Shark the Unknown
Jul 15 at 9:54
Strictly speaking, you might write the second term as $ mathrm e^tACBmathrm e^tB$, but I guess it's for symmetry reasons. Don't forget $B$ and $mathrm e^tB$ commute.
– Bernard
Jul 15 at 9:55
Why do they commute?
– csss
Jul 31 at 15:06