Mixing
Calculating eigen-values and -vectors
$begingroup$
If the vectors $x_1$ and $x_2$ are in the columns of $X$, then what are the eigenvalues and eigenvectors of
$$B = XAX^{-1},$$
where $$A= begin{bmatrix}2&3\0&1end{bmatrix}$$
Is there any way to find the eigenvalues and eigenvectors easier than multiplying the whole matrix and then solving $det(B-lambda I) = 0$?
eigenvalues-eigenvectors
edited Jan 30 at 17:12
J. W. Tanner
4,4051320
asked Jan 30 at 11:36
Gopal ChitaliaGopal Chitalia
153
$endgroup$
add a comment |
$begingroup$
If the vectors $x_1$ and $x_2$ are in the columns of $X$, then what are the eigenvalues and eigenvectors of
$$B = XAX^{-1},$$
where $$A= begin{bmatrix}2&3\0&1end{bmatrix}$$
Is there any way to find the eigenvalues and eigenvectors easier than multiplying the whole matrix and then solving $det(B-lambda I) = 0$?
eigenvalues-eigenvectors
edited Jan 30 at 17:12
J. W. Tanner
4,4051320
asked Jan 30 at 11:36
Gopal ChitaliaGopal Chitalia
153
$endgroup$
add a comment |
$begingroup$
If the vectors $x_1$ and $x_2$ are in the columns of $X$, then what are the eigenvalues and eigenvectors of
$$B = XAX^{-1},$$
where $$A= begin{bmatrix}2&3\0&1end{bmatrix}$$
Is there any way to find the eigenvalues and eigenvectors easier than multiplying the whole matrix and then solving $det(B-lambda I) = 0$?
eigenvalues-eigenvectors
edited Jan 30 at 17:12
J. W. Tanner
4,4051320
asked Jan 30 at 11:36
Gopal ChitaliaGopal Chitalia
153
$endgroup$
If the vectors $x_1$ and $x_2$ are in the columns of $X$, then what are the eigenvalues and eigenvectors of
$$B = XAX^{-1},$$
where $$A= begin{bmatrix}2&3\0&1end{bmatrix}$$
Is there any way to find the eigenvalues and eigenvectors easier than multiplying the whole matrix and then solving $det(B-lambda I) = 0$?
eigenvalues-eigenvectors
eigenvalues-eigenvectors
edited Jan 30 at 17:12
J. W. Tanner
4,4051320
asked Jan 30 at 11:36
Gopal ChitaliaGopal Chitalia
153
edited Jan 30 at 17:12
J. W. Tanner
4,4051320
asked Jan 30 at 11:36
Gopal ChitaliaGopal Chitalia
153
edited Jan 30 at 17:12
J. W. Tanner
4,4051320
edited Jan 30 at 17:12
J. W. Tanner
4,4051320
edited Jan 30 at 17:12
J. W. Tanner
4,4051320
4,4051320
asked Jan 30 at 11:36
Gopal ChitaliaGopal Chitalia
153
asked Jan 30 at 11:36
Gopal ChitaliaGopal Chitalia
153
asked Jan 30 at 11:36
Gopal ChitaliaGopal Chitalia
153
153
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The matrices $A$ and $XAX^{-1}$ are similar and therefore they have the same eigenvalues: $1$ and $2$.
On the other hand, if $v$ is an eigenvector of $A$, then $Xv$ is an eigenvector of $XAX^{-1}$. So, compute the two eigenvectors $v_1$ and $v_2$ of $A$, and then compute $Xv_1$ and $Xv_2$.
edited Jan 30 at 17:40
J. W. Tanner
4,4051320
answered Jan 30 at 11:41
José Carlos SantosJosé Carlos Santos
172k22132239
$endgroup$
$begingroup$
can you please explain to me how $A$ and $XAX^{-1}$ are similar? P.S.:- sorry, if it's a too basic question.
$endgroup$
– Gopal Chitalia
Jan 30 at 11:49
$begingroup$
What do you think “similar” means?
$endgroup$
– José Carlos Santos
Jan 30 at 11:52
$begingroup$
similar as in their eigenvalues are same
$endgroup$
– Gopal Chitalia
Jan 30 at 11:53
1
$begingroup$
No!
$endgroup$
– José Carlos Santos
Jan 30 at 11:53
$begingroup$
ohh! thanks for the info!
$endgroup$
– Gopal Chitalia
Jan 30 at 11:56
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
The matrices $A$ and $XAX^{-1}$ are similar and therefore they have the same eigenvalues: $1$ and $2$.
On the other hand, if $v$ is an eigenvector of $A$, then $Xv$ is an eigenvector of $XAX^{-1}$. So, compute the two eigenvectors $v_1$ and $v_2$ of $A$, and then compute $Xv_1$ and $Xv_2$.
edited Jan 30 at 17:40
J. W. Tanner
4,4051320
answered Jan 30 at 11:41
José Carlos SantosJosé Carlos Santos
172k22132239
$endgroup$
$begingroup$
can you please explain to me how $A$ and $XAX^{-1}$ are similar? P.S.:- sorry, if it's a too basic question.
$endgroup$
– Gopal Chitalia
Jan 30 at 11:49
$begingroup$
What do you think “similar” means?
$endgroup$
– José Carlos Santos
Jan 30 at 11:52
$begingroup$
similar as in their eigenvalues are same
$endgroup$
– Gopal Chitalia
Jan 30 at 11:53
1
$begingroup$
No!
$endgroup$
– José Carlos Santos
Jan 30 at 11:53
$begingroup$
ohh! thanks for the info!
$endgroup$
– Gopal Chitalia
Jan 30 at 11:56
add a comment |
$begingroup$
The matrices $A$ and $XAX^{-1}$ are similar and therefore they have the same eigenvalues: $1$ and $2$.
On the other hand, if $v$ is an eigenvector of $A$, then $Xv$ is an eigenvector of $XAX^{-1}$. So, compute the two eigenvectors $v_1$ and $v_2$ of $A$, and then compute $Xv_1$ and $Xv_2$.
edited Jan 30 at 17:40
J. W. Tanner
4,4051320
answered Jan 30 at 11:41
José Carlos SantosJosé Carlos Santos
172k22132239
$endgroup$
$begingroup$
can you please explain to me how $A$ and $XAX^{-1}$ are similar? P.S.:- sorry, if it's a too basic question.
$endgroup$
– Gopal Chitalia
Jan 30 at 11:49
$begingroup$
What do you think “similar” means?
$endgroup$
– José Carlos Santos
Jan 30 at 11:52
$begingroup$
similar as in their eigenvalues are same
$endgroup$
– Gopal Chitalia
Jan 30 at 11:53
1
$begingroup$
No!
$endgroup$
– José Carlos Santos
Jan 30 at 11:53
$begingroup$
ohh! thanks for the info!
$endgroup$
– Gopal Chitalia
Jan 30 at 11:56
add a comment |
$begingroup$
The matrices $A$ and $XAX^{-1}$ are similar and therefore they have the same eigenvalues: $1$ and $2$.
On the other hand, if $v$ is an eigenvector of $A$, then $Xv$ is an eigenvector of $XAX^{-1}$. So, compute the two eigenvectors $v_1$ and $v_2$ of $A$, and then compute $Xv_1$ and $Xv_2$.
edited Jan 30 at 17:40
J. W. Tanner
4,4051320
answered Jan 30 at 11:41
José Carlos SantosJosé Carlos Santos
172k22132239
$endgroup$
The matrices $A$ and $XAX^{-1}$ are similar and therefore they have the same eigenvalues: $1$ and $2$.
On the other hand, if $v$ is an eigenvector of $A$, then $Xv$ is an eigenvector of $XAX^{-1}$. So, compute the two eigenvectors $v_1$ and $v_2$ of $A$, and then compute $Xv_1$ and $Xv_2$.
edited Jan 30 at 17:40
J. W. Tanner
4,4051320
answered Jan 30 at 11:41
José Carlos SantosJosé Carlos Santos
172k22132239
edited Jan 30 at 17:40
J. W. Tanner
4,4051320
edited Jan 30 at 17:40
J. W. Tanner
4,4051320
edited Jan 30 at 17:40
J. W. Tanner
4,4051320
4,4051320
answered Jan 30 at 11:41
José Carlos SantosJosé Carlos Santos
172k22132239
answered Jan 30 at 11:41
José Carlos SantosJosé Carlos Santos
172k22132239
answered Jan 30 at 11:41
José Carlos SantosJosé Carlos Santos
172k22132239
172k22132239
$begingroup$
can you please explain to me how $A$ and $XAX^{-1}$ are similar? P.S.:- sorry, if it's a too basic question.
$endgroup$
– Gopal Chitalia
Jan 30 at 11:49
$begingroup$
What do you think “similar” means?
$endgroup$
– José Carlos Santos
Jan 30 at 11:52
$begingroup$
similar as in their eigenvalues are same
$endgroup$
– Gopal Chitalia
Jan 30 at 11:53
1
$begingroup$
No!
$endgroup$
– José Carlos Santos
Jan 30 at 11:53
$begingroup$
ohh! thanks for the info!
$endgroup$
– Gopal Chitalia
Jan 30 at 11:56
add a comment |
$begingroup$
can you please explain to me how $A$ and $XAX^{-1}$ are similar? P.S.:- sorry, if it's a too basic question.
$endgroup$
– Gopal Chitalia
Jan 30 at 11:49
$begingroup$
What do you think “similar” means?
$endgroup$
– José Carlos Santos
Jan 30 at 11:52
$begingroup$
similar as in their eigenvalues are same
$endgroup$
– Gopal Chitalia
Jan 30 at 11:53
1
$begingroup$
No!
$endgroup$
– José Carlos Santos
Jan 30 at 11:53
$begingroup$
ohh! thanks for the info!
$endgroup$
– Gopal Chitalia
Jan 30 at 11:56
$begingroup$
can you please explain to me how $A$ and $XAX^{-1}$ are similar? P.S.:- sorry, if it's a too basic question.
$endgroup$
– Gopal Chitalia
Jan 30 at 11:49
$begingroup$
can you please explain to me how $A$ and $XAX^{-1}$ are similar? P.S.:- sorry, if it's a too basic question.
$endgroup$
– Gopal Chitalia
Jan 30 at 11:49
$begingroup$
What do you think “similar” means?
$endgroup$
– José Carlos Santos
Jan 30 at 11:52
$begingroup$
What do you think “similar” means?
$endgroup$
– José Carlos Santos
Jan 30 at 11:52
$begingroup$
similar as in their eigenvalues are same
$endgroup$
– Gopal Chitalia
Jan 30 at 11:53
$begingroup$
similar as in their eigenvalues are same
$endgroup$
– Gopal Chitalia
Jan 30 at 11:53
1
1
$begingroup$
No!
$endgroup$
– José Carlos Santos
Jan 30 at 11:53
$begingroup$
No!
$endgroup$
– José Carlos Santos
Jan 30 at 11:53
$begingroup$
ohh! thanks for the info!
$endgroup$
– Gopal Chitalia
Jan 30 at 11:56
$begingroup$
ohh! thanks for the info!
$endgroup$
– Gopal Chitalia
Jan 30 at 11:56
add a comment |
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