infinite matrix

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I have a question about infinite matrices. Let

beginalign*
A &= beginbmatrix
-q & phspace0.1cmv(1)& 0 &0& dots\
q&-q-phspace0.1cmv(1)&2 phspace0.1cmv(2)&0&dots\
0&q&-q-2 phspace0.1cmv(2)&3 phspace0.1cmv(3)&dots\
0&0&q&-q-3 phspace0.1cmv(3)&dots \
vdots&vdots&vdots&vdots&vdots
endbmatrix
endalign*
which is an infinite matrix but every row and column is finitely supported which means only finite non-zero elements are in every row and column. Now I want to solve ODE : beginequation
partial_t P(t) =A P(t)
labeleq:10
endequation
where $P(t)$ is an infinite vector. Can I say the solution is
beginequation
P(t)=e^tA.P(0)
labeleq:11
endequation
Where
beginequation
e^tA= Sigma_k=0^infty fract^k A^kk!
labeleq:12
endequation
I know that above solution is valid when $A$ is finite but I'm not really sure for infinite matrices. I'd appreciate it if anyone could help me with this.




matrices differential-equations




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  • Perhaps Differential equations, infinite-order system of ?
    – user539887
    Aug 3 at 13:44















up vote
2
down vote

favorite
1












I have a question about infinite matrices. Let

beginalign*
A &= beginbmatrix
-q & phspace0.1cmv(1)& 0 &0& dots\
q&-q-phspace0.1cmv(1)&2 phspace0.1cmv(2)&0&dots\
0&q&-q-2 phspace0.1cmv(2)&3 phspace0.1cmv(3)&dots\
0&0&q&-q-3 phspace0.1cmv(3)&dots \
vdots&vdots&vdots&vdots&vdots
endbmatrix
endalign*
which is an infinite matrix but every row and column is finitely supported which means only finite non-zero elements are in every row and column. Now I want to solve ODE : beginequation
partial_t P(t) =A P(t)
labeleq:10
endequation
where $P(t)$ is an infinite vector. Can I say the solution is
beginequation
P(t)=e^tA.P(0)
labeleq:11
endequation
Where
beginequation
e^tA= Sigma_k=0^infty fract^k A^kk!
labeleq:12
endequation
I know that above solution is valid when $A$ is finite but I'm not really sure for infinite matrices. I'd appreciate it if anyone could help me with this.




matrices differential-equations




share|cite|improve this question





















  • Perhaps Differential equations, infinite-order system of ?
    – user539887
    Aug 3 at 13:44













up vote
2
down vote

favorite
1









up vote
2
down vote

favorite
1






1





I have a question about infinite matrices. Let

beginalign*
A &= beginbmatrix
-q & phspace0.1cmv(1)& 0 &0& dots\
q&-q-phspace0.1cmv(1)&2 phspace0.1cmv(2)&0&dots\
0&q&-q-2 phspace0.1cmv(2)&3 phspace0.1cmv(3)&dots\
0&0&q&-q-3 phspace0.1cmv(3)&dots \
vdots&vdots&vdots&vdots&vdots
endbmatrix
endalign*
which is an infinite matrix but every row and column is finitely supported which means only finite non-zero elements are in every row and column. Now I want to solve ODE : beginequation
partial_t P(t) =A P(t)
labeleq:10
endequation
where $P(t)$ is an infinite vector. Can I say the solution is
beginequation
P(t)=e^tA.P(0)
labeleq:11
endequation
Where
beginequation
e^tA= Sigma_k=0^infty fract^k A^kk!
labeleq:12
endequation
I know that above solution is valid when $A$ is finite but I'm not really sure for infinite matrices. I'd appreciate it if anyone could help me with this.




matrices differential-equations




share|cite|improve this question













I have a question about infinite matrices. Let

beginalign*
A &= beginbmatrix
-q & phspace0.1cmv(1)& 0 &0& dots\
q&-q-phspace0.1cmv(1)&2 phspace0.1cmv(2)&0&dots\
0&q&-q-2 phspace0.1cmv(2)&3 phspace0.1cmv(3)&dots\
0&0&q&-q-3 phspace0.1cmv(3)&dots \
vdots&vdots&vdots&vdots&vdots
endbmatrix
endalign*
which is an infinite matrix but every row and column is finitely supported which means only finite non-zero elements are in every row and column. Now I want to solve ODE : beginequation
partial_t P(t) =A P(t)
labeleq:10
endequation
where $P(t)$ is an infinite vector. Can I say the solution is
beginequation
P(t)=e^tA.P(0)
labeleq:11
endequation
Where
beginequation
e^tA= Sigma_k=0^infty fract^k A^kk!
labeleq:12
endequation
I know that above solution is valid when $A$ is finite but I'm not really sure for infinite matrices. I'd appreciate it if anyone could help me with this.





matrices differential-equations





share|cite|improve this question












share|cite|improve this question




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edited Aug 3 at 3:35
























asked Aug 3 at 3:21









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  • Perhaps Differential equations, infinite-order system of ?
    – user539887
    Aug 3 at 13:44

















  • Perhaps Differential equations, infinite-order system of ?
    – user539887
    Aug 3 at 13:44
















Perhaps Differential equations, infinite-order system of ?
– user539887
Aug 3 at 13:44





Perhaps Differential equations, infinite-order system of ?
– user539887
Aug 3 at 13:44
















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