Mixing
Finding all subspaces invariant under F.(2)
$begingroup$
I have a question regarding the question in the link here:
Finding all subspaces invariant under F.
The answer is so convincing in the above link but I just want to know why the matrix corresponding to the operator $alpha$ will be diagonal, is there a theorem saying this?
EDIT:
Also I am wondering how is the answer mentioned in the above link by @Berci leads to the answer mentioned in the OP question in the above link (The answer at the back of the book)
Thanks!
linear-algebra representation-theory invariant-theory invariant-subspace
asked Jan 30 at 16:59
hopefullyhopefully
270215
$endgroup$
add a comment |
$begingroup$
I have a question regarding the question in the link here:
Finding all subspaces invariant under F.
The answer is so convincing in the above link but I just want to know why the matrix corresponding to the operator $alpha$ will be diagonal, is there a theorem saying this?
EDIT:
Also I am wondering how is the answer mentioned in the above link by @Berci leads to the answer mentioned in the OP question in the above link (The answer at the back of the book)
Thanks!
linear-algebra representation-theory invariant-theory invariant-subspace
asked Jan 30 at 16:59
hopefullyhopefully
270215
$endgroup$
$begingroup$
The matrix of a linear transformation is diagonal w.r.t. basis $(v_i)$ iff each basis vector $v_i$ is an eigenvector.
$endgroup$
– Berci
Jan 30 at 18:34
$begingroup$
is this a theorem@Berci?
$endgroup$
– hopefully
Jan 30 at 18:41
1
$begingroup$
Well, this easily follows from the definition of a matrix of a linear map w.r.t. some basis. Nevertheless, we can call it a theorem.
$endgroup$
– Berci
Jan 30 at 18:43
$begingroup$
Can you tell me a book where I could find this theorem in please? @Berci
$endgroup$
– hopefully
Jan 30 at 19:12
$begingroup$
Thank you I found it @Berci
$endgroup$
– hopefully
Jan 30 at 19:16
add a comment |
$begingroup$
I have a question regarding the question in the link here:
Finding all subspaces invariant under F.
The answer is so convincing in the above link but I just want to know why the matrix corresponding to the operator $alpha$ will be diagonal, is there a theorem saying this?
EDIT:
Also I am wondering how is the answer mentioned in the above link by @Berci leads to the answer mentioned in the OP question in the above link (The answer at the back of the book)
Thanks!
linear-algebra representation-theory invariant-theory invariant-subspace
asked Jan 30 at 16:59
hopefullyhopefully
270215
$endgroup$
I have a question regarding the question in the link here:
Finding all subspaces invariant under F.
The answer is so convincing in the above link but I just want to know why the matrix corresponding to the operator $alpha$ will be diagonal, is there a theorem saying this?
EDIT:
Also I am wondering how is the answer mentioned in the above link by @Berci leads to the answer mentioned in the OP question in the above link (The answer at the back of the book)
Thanks!
linear-algebra representation-theory invariant-theory invariant-subspace
linear-algebra representation-theory invariant-theory invariant-subspace
asked Jan 30 at 16:59
hopefullyhopefully
270215
asked Jan 30 at 16:59
hopefullyhopefully
270215
edited Jan 31 at 2:20
hopefully
asked Jan 30 at 16:59
hopefullyhopefully
270215
asked Jan 30 at 16:59
hopefullyhopefully
270215
asked Jan 30 at 16:59
hopefullyhopefully
270215
270215
$begingroup$
The matrix of a linear transformation is diagonal w.r.t. basis $(v_i)$ iff each basis vector $v_i$ is an eigenvector.
$endgroup$
– Berci
Jan 30 at 18:34
$begingroup$
is this a theorem@Berci?
$endgroup$
– hopefully
Jan 30 at 18:41
1
$begingroup$
Well, this easily follows from the definition of a matrix of a linear map w.r.t. some basis. Nevertheless, we can call it a theorem.
$endgroup$
– Berci
Jan 30 at 18:43
$begingroup$
Can you tell me a book where I could find this theorem in please? @Berci
$endgroup$
– hopefully
Jan 30 at 19:12
$begingroup$
Thank you I found it @Berci
$endgroup$
– hopefully
Jan 30 at 19:16
add a comment |
$begingroup$
The matrix of a linear transformation is diagonal w.r.t. basis $(v_i)$ iff each basis vector $v_i$ is an eigenvector.
$endgroup$
– Berci
Jan 30 at 18:34
$begingroup$
is this a theorem@Berci?
$endgroup$
– hopefully
Jan 30 at 18:41
1
$begingroup$
Well, this easily follows from the definition of a matrix of a linear map w.r.t. some basis. Nevertheless, we can call it a theorem.
$endgroup$
– Berci
Jan 30 at 18:43
$begingroup$
Can you tell me a book where I could find this theorem in please? @Berci
$endgroup$
– hopefully
Jan 30 at 19:12
$begingroup$
Thank you I found it @Berci
$endgroup$
– hopefully
Jan 30 at 19:16
$begingroup$
The matrix of a linear transformation is diagonal w.r.t. basis $(v_i)$ iff each basis vector $v_i$ is an eigenvector.
$endgroup$
– Berci
Jan 30 at 18:34
$begingroup$
The matrix of a linear transformation is diagonal w.r.t. basis $(v_i)$ iff each basis vector $v_i$ is an eigenvector.
$endgroup$
– Berci
Jan 30 at 18:34
$begingroup$
is this a theorem@Berci?
$endgroup$
– hopefully
Jan 30 at 18:41
$begingroup$
is this a theorem@Berci?
$endgroup$
– hopefully
Jan 30 at 18:41
1
1
$begingroup$
Well, this easily follows from the definition of a matrix of a linear map w.r.t. some basis. Nevertheless, we can call it a theorem.
$endgroup$
– Berci
Jan 30 at 18:43
$begingroup$
Well, this easily follows from the definition of a matrix of a linear map w.r.t. some basis. Nevertheless, we can call it a theorem.
$endgroup$
– Berci
Jan 30 at 18:43
$begingroup$
Can you tell me a book where I could find this theorem in please? @Berci
$endgroup$
– hopefully
Jan 30 at 19:12
$begingroup$
Can you tell me a book where I could find this theorem in please? @Berci
$endgroup$
– hopefully
Jan 30 at 19:12
$begingroup$
Thank you I found it @Berci
$endgroup$
– hopefully
Jan 30 at 19:16
$begingroup$
Thank you I found it @Berci
$endgroup$
– hopefully
Jan 30 at 19:16
add a comment |
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$begingroup$
The matrix of a linear transformation is diagonal w.r.t. basis $(v_i)$ iff each basis vector $v_i$ is an eigenvector.
$endgroup$
– Berci
Jan 30 at 18:34
$begingroup$
is this a theorem@Berci?
$endgroup$
– hopefully
Jan 30 at 18:41
1
$begingroup$
Well, this easily follows from the definition of a matrix of a linear map w.r.t. some basis. Nevertheless, we can call it a theorem.
$endgroup$
– Berci
Jan 30 at 18:43
$begingroup$
Can you tell me a book where I could find this theorem in please? @Berci
$endgroup$
– hopefully
Jan 30 at 19:12
$begingroup$
Thank you I found it @Berci
$endgroup$
– hopefully
Jan 30 at 19:16