Mixing
Changing a complex number series to a matrix series.
$begingroup$
I'd like to know if the following assertion is true.
Say you have a converging complex series: $sum_{n=0}^{infty}f_n(z)=g(z)$. Then, if you choose a complex matrix $Z$ such that $sum_{n=0}^{infty}f_n(Z)$ converges (simply), then it must converge to $g(Z)$. If there are any constants $alpha_k$ in $g(z)$ you replace them by $alpha_k I$, where $I$ is the identity matrix.
I'm asking this because I believe it is being implicitly used in a proof my professor has given for the fact that every non-singular matrix $C$ can be written as $e^B$, for some $B$. I don't find it intuitive at all.
Thanks in advance for any help.
sequences-and-series matrices convergence
asked Jan 30 at 14:11
Leo BiancoLeo Bianco
308212
$endgroup$
add a comment |
$begingroup$
I'd like to know if the following assertion is true.
Say you have a converging complex series: $sum_{n=0}^{infty}f_n(z)=g(z)$. Then, if you choose a complex matrix $Z$ such that $sum_{n=0}^{infty}f_n(Z)$ converges (simply), then it must converge to $g(Z)$. If there are any constants $alpha_k$ in $g(z)$ you replace them by $alpha_k I$, where $I$ is the identity matrix.
I'm asking this because I believe it is being implicitly used in a proof my professor has given for the fact that every non-singular matrix $C$ can be written as $e^B$, for some $B$. I don't find it intuitive at all.
Thanks in advance for any help.
sequences-and-series matrices convergence
asked Jan 30 at 14:11
Leo BiancoLeo Bianco
308212
$endgroup$
1
$begingroup$
Given a function $g$ over the complex numbers, how exactly do you (or how did your professor) define $g(Z)$ for a matrix $Z$? There are different approaches here, and which one we start with matters.
$endgroup$
– Omnomnomnom
Jan 30 at 14:25
add a comment |
$begingroup$
I'd like to know if the following assertion is true.
Say you have a converging complex series: $sum_{n=0}^{infty}f_n(z)=g(z)$. Then, if you choose a complex matrix $Z$ such that $sum_{n=0}^{infty}f_n(Z)$ converges (simply), then it must converge to $g(Z)$. If there are any constants $alpha_k$ in $g(z)$ you replace them by $alpha_k I$, where $I$ is the identity matrix.
I'm asking this because I believe it is being implicitly used in a proof my professor has given for the fact that every non-singular matrix $C$ can be written as $e^B$, for some $B$. I don't find it intuitive at all.
Thanks in advance for any help.
sequences-and-series matrices convergence
asked Jan 30 at 14:11
Leo BiancoLeo Bianco
308212
$endgroup$
I'd like to know if the following assertion is true.
Say you have a converging complex series: $sum_{n=0}^{infty}f_n(z)=g(z)$. Then, if you choose a complex matrix $Z$ such that $sum_{n=0}^{infty}f_n(Z)$ converges (simply), then it must converge to $g(Z)$. If there are any constants $alpha_k$ in $g(z)$ you replace them by $alpha_k I$, where $I$ is the identity matrix.
I'm asking this because I believe it is being implicitly used in a proof my professor has given for the fact that every non-singular matrix $C$ can be written as $e^B$, for some $B$. I don't find it intuitive at all.
Thanks in advance for any help.
sequences-and-series matrices convergence
sequences-and-series matrices convergence
asked Jan 30 at 14:11
Leo BiancoLeo Bianco
308212
asked Jan 30 at 14:11
Leo BiancoLeo Bianco
308212
asked Jan 30 at 14:11
Leo BiancoLeo Bianco
308212
asked Jan 30 at 14:11
Leo BiancoLeo Bianco
308212
asked Jan 30 at 14:11
Leo BiancoLeo Bianco
308212
308212
1
$begingroup$
Given a function $g$ over the complex numbers, how exactly do you (or how did your professor) define $g(Z)$ for a matrix $Z$? There are different approaches here, and which one we start with matters.
$endgroup$
– Omnomnomnom
Jan 30 at 14:25
add a comment |
1
$begingroup$
Given a function $g$ over the complex numbers, how exactly do you (or how did your professor) define $g(Z)$ for a matrix $Z$? There are different approaches here, and which one we start with matters.
$endgroup$
– Omnomnomnom
Jan 30 at 14:25
1
1
$begingroup$
Given a function $g$ over the complex numbers, how exactly do you (or how did your professor) define $g(Z)$ for a matrix $Z$? There are different approaches here, and which one we start with matters.
$endgroup$
– Omnomnomnom
Jan 30 at 14:25
$begingroup$
Given a function $g$ over the complex numbers, how exactly do you (or how did your professor) define $g(Z)$ for a matrix $Z$? There are different approaches here, and which one we start with matters.
$endgroup$
– Omnomnomnom
Jan 30 at 14:25
add a comment |
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$begingroup$
Given a function $g$ over the complex numbers, how exactly do you (or how did your professor) define $g(Z)$ for a matrix $Z$? There are different approaches here, and which one we start with matters.
$endgroup$
– Omnomnomnom
Jan 30 at 14:25