Show that a matrix is diagonal [duplicate]
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This question already has an answer here:
Simultaneous diagonalization
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My task given by my professor was the following:
$Let ,, A, ,, B in mathbb{C}^{ntimes n}$ be selfadjoint and such that $[A, B] := AB-BA=0.$ Show that $C:= A+iB$ is normal. Show further that there is a unitary $U in mathbb{C}^{ntimes n} ; text{such that} ; U^*AU ; text{and} ; U^*BU$ are both diagonal.
I have proven that $C$ is normal but I'm having problems with proving that the terms are both diagonal matricies.
linear-algebra proof-writing diagonalization
edited Jan 8 at 17:45
Robert Lewis
48.6k23167
asked Jan 8 at 17:38
Fo Young Areal LoFo Young Areal Lo
345
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marked as duplicate by Acccumulation, Alexander Gruber♦ Jan 8 at 22:13
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
Simultaneous diagonalization
2 answers
My task given by my professor was the following:
$Let ,, A, ,, B in mathbb{C}^{ntimes n}$ be selfadjoint and such that $[A, B] := AB-BA=0.$ Show that $C:= A+iB$ is normal. Show further that there is a unitary $U in mathbb{C}^{ntimes n} ; text{such that} ; U^*AU ; text{and} ; U^*BU$ are both diagonal.
I have proven that $C$ is normal but I'm having problems with proving that the terms are both diagonal matricies.
linear-algebra proof-writing diagonalization
edited Jan 8 at 17:45
Robert Lewis
48.6k23167
asked Jan 8 at 17:38
Fo Young Areal LoFo Young Areal Lo
345
$endgroup$
marked as duplicate by Acccumulation, Alexander Gruber♦ Jan 8 at 22:13
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
$begingroup$
This question already has an answer here:
Simultaneous diagonalization
2 answers
My task given by my professor was the following:
$Let ,, A, ,, B in mathbb{C}^{ntimes n}$ be selfadjoint and such that $[A, B] := AB-BA=0.$ Show that $C:= A+iB$ is normal. Show further that there is a unitary $U in mathbb{C}^{ntimes n} ; text{such that} ; U^*AU ; text{and} ; U^*BU$ are both diagonal.
I have proven that $C$ is normal but I'm having problems with proving that the terms are both diagonal matricies.
linear-algebra proof-writing diagonalization
edited Jan 8 at 17:45
Robert Lewis
48.6k23167
asked Jan 8 at 17:38
Fo Young Areal LoFo Young Areal Lo
345
$endgroup$
This question already has an answer here:
Simultaneous diagonalization
2 answers
My task given by my professor was the following:
$Let ,, A, ,, B in mathbb{C}^{ntimes n}$ be selfadjoint and such that $[A, B] := AB-BA=0.$ Show that $C:= A+iB$ is normal. Show further that there is a unitary $U in mathbb{C}^{ntimes n} ; text{such that} ; U^*AU ; text{and} ; U^*BU$ are both diagonal.
I have proven that $C$ is normal but I'm having problems with proving that the terms are both diagonal matricies.
This question already has an answer here:
Simultaneous diagonalization
2 answers
linear-algebra proof-writing diagonalization
linear-algebra proof-writing diagonalization
edited Jan 8 at 17:45
Robert Lewis
48.6k23167
asked Jan 8 at 17:38
Fo Young Areal LoFo Young Areal Lo
345
edited Jan 8 at 17:45
Robert Lewis
48.6k23167
asked Jan 8 at 17:38
Fo Young Areal LoFo Young Areal Lo
345
edited Jan 8 at 17:45
Robert Lewis
48.6k23167
edited Jan 8 at 17:45
Robert Lewis
48.6k23167
edited Jan 8 at 17:45
Robert Lewis
48.6k23167
48.6k23167
asked Jan 8 at 17:38
Fo Young Areal LoFo Young Areal Lo
345
asked Jan 8 at 17:38
Fo Young Areal LoFo Young Areal Lo
345
asked Jan 8 at 17:38
Fo Young Areal LoFo Young Areal Lo
345
345
marked as duplicate by Acccumulation, Alexander Gruber♦ Jan 8 at 22:13
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Acccumulation, Alexander Gruber♦ Jan 8 at 22:13
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
add a comment |
add a comment |
1 Answer
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Hint: If $U^* C U = D$ is diagonal, $U^* C^* U = (U^* C U)^* = D^*$ which is also diagonal. Now what are $(C + C^*)/2$ and $(C - C^*)/(2i)$?
answered Jan 8 at 17:56
Robert IsraelRobert Israel
330k23219473
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I don't really know how to use your hint for my proof. Can you give me a deeper explanation?
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– Fo Young Areal Lo
Jan 9 at 15:42
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Thank you after thinking about it I got it. I'm fine now.
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– Fo Young Areal Lo
Jan 9 at 16:36
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: If $U^* C U = D$ is diagonal, $U^* C^* U = (U^* C U)^* = D^*$ which is also diagonal. Now what are $(C + C^*)/2$ and $(C - C^*)/(2i)$?
answered Jan 8 at 17:56
Robert IsraelRobert Israel
330k23219473
$endgroup$
$begingroup$
I don't really know how to use your hint for my proof. Can you give me a deeper explanation?
$endgroup$
– Fo Young Areal Lo
Jan 9 at 15:42
$begingroup$
Thank you after thinking about it I got it. I'm fine now.
$endgroup$
– Fo Young Areal Lo
Jan 9 at 16:36
add a comment |
$begingroup$
Hint: If $U^* C U = D$ is diagonal, $U^* C^* U = (U^* C U)^* = D^*$ which is also diagonal. Now what are $(C + C^*)/2$ and $(C - C^*)/(2i)$?
answered Jan 8 at 17:56
Robert IsraelRobert Israel
330k23219473
$endgroup$
$begingroup$
I don't really know how to use your hint for my proof. Can you give me a deeper explanation?
$endgroup$
– Fo Young Areal Lo
Jan 9 at 15:42
$begingroup$
Thank you after thinking about it I got it. I'm fine now.
$endgroup$
– Fo Young Areal Lo
Jan 9 at 16:36
add a comment |
$begingroup$
Hint: If $U^* C U = D$ is diagonal, $U^* C^* U = (U^* C U)^* = D^*$ which is also diagonal. Now what are $(C + C^*)/2$ and $(C - C^*)/(2i)$?
answered Jan 8 at 17:56
Robert IsraelRobert Israel
330k23219473
$endgroup$
Hint: If $U^* C U = D$ is diagonal, $U^* C^* U = (U^* C U)^* = D^*$ which is also diagonal. Now what are $(C + C^*)/2$ and $(C - C^*)/(2i)$?
answered Jan 8 at 17:56
Robert IsraelRobert Israel
330k23219473
answered Jan 8 at 17:56
Robert IsraelRobert Israel
330k23219473
answered Jan 8 at 17:56
Robert IsraelRobert Israel
330k23219473
answered Jan 8 at 17:56
Robert IsraelRobert Israel
330k23219473
330k23219473
$begingroup$
I don't really know how to use your hint for my proof. Can you give me a deeper explanation?
$endgroup$
– Fo Young Areal Lo
Jan 9 at 15:42
$begingroup$
Thank you after thinking about it I got it. I'm fine now.
$endgroup$
– Fo Young Areal Lo
Jan 9 at 16:36
add a comment |
$begingroup$
I don't really know how to use your hint for my proof. Can you give me a deeper explanation?
$endgroup$
– Fo Young Areal Lo
Jan 9 at 15:42
$begingroup$
Thank you after thinking about it I got it. I'm fine now.
$endgroup$
– Fo Young Areal Lo
Jan 9 at 16:36
$begingroup$
I don't really know how to use your hint for my proof. Can you give me a deeper explanation?
$endgroup$
– Fo Young Areal Lo
Jan 9 at 15:42
$begingroup$
I don't really know how to use your hint for my proof. Can you give me a deeper explanation?
$endgroup$
– Fo Young Areal Lo
Jan 9 at 15:42
$begingroup$
Thank you after thinking about it I got it. I'm fine now.
$endgroup$
– Fo Young Areal Lo
Jan 9 at 16:36
$begingroup$
Thank you after thinking about it I got it. I'm fine now.
$endgroup$
– Fo Young Areal Lo
Jan 9 at 16:36
add a comment |
