Differentiability of $g(t) = f(t)^-1$, where $f : I subset mathbbR to mathrmGL(mathbbR^n)$ is differentiable [closed]
Clash Royale CLAN TAG#URR8PPP
up vote
-1
down vote
favorite
Let $Isubset mathbbR$ be an open set and $f:I to mathrmGL(mathbbR^n)$ differentiable. Show that $g:Ito mathrmGL(mathbbR^n)$, defined by $g(t)=f(t)^-1$, is differentiable and calculate $g'$.
I have tried some calculations, but I can't prove.
real-analysis matrices analysis
closed as off-topic by Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy Aug 22 at 10:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy
add a comment |Â
up vote
-1
down vote
favorite
Let $Isubset mathbbR$ be an open set and $f:I to mathrmGL(mathbbR^n)$ differentiable. Show that $g:Ito mathrmGL(mathbbR^n)$, defined by $g(t)=f(t)^-1$, is differentiable and calculate $g'$.
I have tried some calculations, but I can't prove.
real-analysis matrices analysis
closed as off-topic by Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy Aug 22 at 10:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy
1
What did you try?
â Shaun
Aug 21 at 21:26
@Shaun I have written g(t+h)-g(t) and tried to related with f. :(
â 7697
Aug 21 at 21:34
Do you know how to write down $A^-1$ in terms of $A$?
â John Ma
Aug 21 at 21:42
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Let $Isubset mathbbR$ be an open set and $f:I to mathrmGL(mathbbR^n)$ differentiable. Show that $g:Ito mathrmGL(mathbbR^n)$, defined by $g(t)=f(t)^-1$, is differentiable and calculate $g'$.
I have tried some calculations, but I can't prove.
real-analysis matrices analysis
Let $Isubset mathbbR$ be an open set and $f:I to mathrmGL(mathbbR^n)$ differentiable. Show that $g:Ito mathrmGL(mathbbR^n)$, defined by $g(t)=f(t)^-1$, is differentiable and calculate $g'$.
I have tried some calculations, but I can't prove.
real-analysis matrices analysis
edited Aug 22 at 9:53
Brahadeesh
4,13631550
4,13631550
asked Aug 21 at 21:14
7697
2489
2489
closed as off-topic by Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy Aug 22 at 10:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy
closed as off-topic by Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy Aug 22 at 10:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." â Shaun, John Ma, Xander Henderson, Taroccoesbrocco, amWhy
1
What did you try?
â Shaun
Aug 21 at 21:26
@Shaun I have written g(t+h)-g(t) and tried to related with f. :(
â 7697
Aug 21 at 21:34
Do you know how to write down $A^-1$ in terms of $A$?
â John Ma
Aug 21 at 21:42
add a comment |Â
1
What did you try?
â Shaun
Aug 21 at 21:26
@Shaun I have written g(t+h)-g(t) and tried to related with f. :(
â 7697
Aug 21 at 21:34
Do you know how to write down $A^-1$ in terms of $A$?
â John Ma
Aug 21 at 21:42
1
1
What did you try?
â Shaun
Aug 21 at 21:26
What did you try?
â Shaun
Aug 21 at 21:26
@Shaun I have written g(t+h)-g(t) and tried to related with f. :(
â 7697
Aug 21 at 21:34
@Shaun I have written g(t+h)-g(t) and tried to related with f. :(
â 7697
Aug 21 at 21:34
Do you know how to write down $A^-1$ in terms of $A$?
â John Ma
Aug 21 at 21:42
Do you know how to write down $A^-1$ in terms of $A$?
â John Ma
Aug 21 at 21:42
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
$g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.
To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.
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
$g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.
To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.
add a comment |Â
up vote
1
down vote
accepted
$g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.
To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
$g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.
To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.
$g(t) = f(t)^-1$ is differentiable because $f(t)$ is differentiable and each entry of $f(t)^-1$ is a rational function of the entries of $f(t)$.
To compute the derivative of $g$, use $mathrmId = f(t) g(t)$ for all $t in I$ and then apply the chain rule.
answered Aug 22 at 9:50
Brahadeesh
4,13631550
4,13631550
add a comment |Â
add a comment |Â
1
What did you try?
â Shaun
Aug 21 at 21:26
@Shaun I have written g(t+h)-g(t) and tried to related with f. :(
â 7697
Aug 21 at 21:34
Do you know how to write down $A^-1$ in terms of $A$?
â John Ma
Aug 21 at 21:42