Does Cayley Hamilton Theorem apply for non-diagonalizable matrices as well?
up vote
2
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Cayley Hamilton Theorem says that a matrix $A$ satisfies its characteristic equation. My professor proved this for diagonalizable matrices. What happens if $A$ is not diagonalizable? Does the C-H Theorem still hold? Can you give a proof why it holds or why it does not?
Thanks!
linear-algebra matrices characteristic-functions cayley-hamilton
asked Nov 2 at 2:21
Nagabhushan S N
255114
add a comment |
up vote
2
down vote
favorite
Cayley Hamilton Theorem says that a matrix $A$ satisfies its characteristic equation. My professor proved this for diagonalizable matrices. What happens if $A$ is not diagonalizable? Does the C-H Theorem still hold? Can you give a proof why it holds or why it does not?
Thanks!
linear-algebra matrices characteristic-functions cayley-hamilton
asked Nov 2 at 2:21
Nagabhushan S N
255114
It does indeed hold in general
– RhythmInk
Nov 2 at 2:22
1
Yes, it still holds. One way to see this is to use the fact that diagonalizable matrices are dense in all matrices.
– Qiaochu Yuan
Nov 2 at 2:22
@QiaochuYuan, can you please elaborate?
– Nagabhushan S N
Nov 2 at 2:43
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Cayley Hamilton Theorem says that a matrix $A$ satisfies its characteristic equation. My professor proved this for diagonalizable matrices. What happens if $A$ is not diagonalizable? Does the C-H Theorem still hold? Can you give a proof why it holds or why it does not?
Thanks!
linear-algebra matrices characteristic-functions cayley-hamilton
asked Nov 2 at 2:21
Nagabhushan S N
255114
Cayley Hamilton Theorem says that a matrix $A$ satisfies its characteristic equation. My professor proved this for diagonalizable matrices. What happens if $A$ is not diagonalizable? Does the C-H Theorem still hold? Can you give a proof why it holds or why it does not?
Thanks!
linear-algebra matrices characteristic-functions cayley-hamilton
linear-algebra matrices characteristic-functions cayley-hamilton
asked Nov 2 at 2:21
Nagabhushan S N
255114
asked Nov 2 at 2:21
Nagabhushan S N
255114
asked Nov 2 at 2:21
Nagabhushan S N
255114
asked Nov 2 at 2:21
Nagabhushan S N
255114
asked Nov 2 at 2:21
Nagabhushan S N
255114
255114
It does indeed hold in general
– RhythmInk
Nov 2 at 2:22
1
Yes, it still holds. One way to see this is to use the fact that diagonalizable matrices are dense in all matrices.
– Qiaochu Yuan
Nov 2 at 2:22
@QiaochuYuan, can you please elaborate?
– Nagabhushan S N
Nov 2 at 2:43
add a comment |
It does indeed hold in general
– RhythmInk
Nov 2 at 2:22
1
Yes, it still holds. One way to see this is to use the fact that diagonalizable matrices are dense in all matrices.
– Qiaochu Yuan
Nov 2 at 2:22
@QiaochuYuan, can you please elaborate?
– Nagabhushan S N
Nov 2 at 2:43
It does indeed hold in general
– RhythmInk
Nov 2 at 2:22
It does indeed hold in general
– RhythmInk
Nov 2 at 2:22
1
1
Yes, it still holds. One way to see this is to use the fact that diagonalizable matrices are dense in all matrices.
– Qiaochu Yuan
Nov 2 at 2:22
Yes, it still holds. One way to see this is to use the fact that diagonalizable matrices are dense in all matrices.
– Qiaochu Yuan
Nov 2 at 2:22
@QiaochuYuan, can you please elaborate?
– Nagabhushan S N
Nov 2 at 2:43
@QiaochuYuan, can you please elaborate?
– Nagabhushan S N
Nov 2 at 2:43
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
It holds in general. Also the proof can be found in https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem
answered Nov 24 at 17:08
Mostafa Ayaz
13.4k3836
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
It holds in general. Also the proof can be found in https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem
answered Nov 24 at 17:08
Mostafa Ayaz
13.4k3836
add a comment |
up vote
1
down vote
accepted
It holds in general. Also the proof can be found in https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem
answered Nov 24 at 17:08
Mostafa Ayaz
13.4k3836
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
It holds in general. Also the proof can be found in https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem
answered Nov 24 at 17:08
Mostafa Ayaz
13.4k3836
It holds in general. Also the proof can be found in https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem
answered Nov 24 at 17:08
Mostafa Ayaz
13.4k3836
answered Nov 24 at 17:08
Mostafa Ayaz
13.4k3836
answered Nov 24 at 17:08
Mostafa Ayaz
13.4k3836
answered Nov 24 at 17:08
Mostafa Ayaz
13.4k3836
13.4k3836
add a comment |
add a comment |
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It does indeed hold in general
– RhythmInk
Nov 2 at 2:22
1
Yes, it still holds. One way to see this is to use the fact that diagonalizable matrices are dense in all matrices.
– Qiaochu Yuan
Nov 2 at 2:22
@QiaochuYuan, can you please elaborate?
– Nagabhushan S N
Nov 2 at 2:43