Relation between simultaneously diagonalizable and eigenspace
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A and B are two diagonalizable matrices with real coefficients.
1) Prove that if A, B are 2x2 matrices, then they are simultaneously diagonalizable if and only if they have the same eigenspaces or one of the two matrices is multiple to the identity.
2) Prove that this is not true for 3x3 matrices.
I am not able to solve it. Have you any suggestions on how to proceed?
matrices eigenvalues-eigenvectors diagonalization
asked Sep 5 at 8:30
Phi_24
675
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up vote
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favorite
A and B are two diagonalizable matrices with real coefficients.
1) Prove that if A, B are 2x2 matrices, then they are simultaneously diagonalizable if and only if they have the same eigenspaces or one of the two matrices is multiple to the identity.
2) Prove that this is not true for 3x3 matrices.
I am not able to solve it. Have you any suggestions on how to proceed?
matrices eigenvalues-eigenvectors diagonalization
asked Sep 5 at 8:30
Phi_24
675
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
A and B are two diagonalizable matrices with real coefficients.
1) Prove that if A, B are 2x2 matrices, then they are simultaneously diagonalizable if and only if they have the same eigenspaces or one of the two matrices is multiple to the identity.
2) Prove that this is not true for 3x3 matrices.
I am not able to solve it. Have you any suggestions on how to proceed?
matrices eigenvalues-eigenvectors diagonalization
asked Sep 5 at 8:30
Phi_24
675
A and B are two diagonalizable matrices with real coefficients.
1) Prove that if A, B are 2x2 matrices, then they are simultaneously diagonalizable if and only if they have the same eigenspaces or one of the two matrices is multiple to the identity.
2) Prove that this is not true for 3x3 matrices.
I am not able to solve it. Have you any suggestions on how to proceed?
matrices eigenvalues-eigenvectors diagonalization
matrices eigenvalues-eigenvectors diagonalization
asked Sep 5 at 8:30
Phi_24
675
asked Sep 5 at 8:30
Phi_24
675
asked Sep 5 at 8:30
Phi_24
675
asked Sep 5 at 8:30
Phi_24
675
asked Sep 5 at 8:30
Phi_24
675
675
add a comment |Â
add a comment |Â
1 Answer
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If one of the matrices is a multiple of the identity then we have simultaneous diagonalizability. The same is true if they share the same eigenspaces.
Now let $A$ and $B$ be simultaneously diagonalizable. Then there is a basis $(x_1,x_2)$ of vectors that are eigenvectors for both matrices. If one of the matrices is a multiple of the identity there is nothing to prove. If this is not the case then both matrices have two different eigenvalues with eigenspaces of dimension one. And the eigenspaces coincide.
For the $3times 3$ case, try to construct to matrices with a basis $(x_1,x_2,x_3)$ of common eigenvectors such that one matrix has two eigenspaces spanned by $x_1,x_2$ and $x_3$, the other has two eigenspaces spanned by $x_1$ and $x_2,x_3$.
answered Sep 5 at 14:16
daw
22.2k1542
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
accepted
If one of the matrices is a multiple of the identity then we have simultaneous diagonalizability. The same is true if they share the same eigenspaces.
Now let $A$ and $B$ be simultaneously diagonalizable. Then there is a basis $(x_1,x_2)$ of vectors that are eigenvectors for both matrices. If one of the matrices is a multiple of the identity there is nothing to prove. If this is not the case then both matrices have two different eigenvalues with eigenspaces of dimension one. And the eigenspaces coincide.
For the $3times 3$ case, try to construct to matrices with a basis $(x_1,x_2,x_3)$ of common eigenvectors such that one matrix has two eigenspaces spanned by $x_1,x_2$ and $x_3$, the other has two eigenspaces spanned by $x_1$ and $x_2,x_3$.
answered Sep 5 at 14:16
daw
22.2k1542
add a comment |Â
up vote
0
down vote
accepted
If one of the matrices is a multiple of the identity then we have simultaneous diagonalizability. The same is true if they share the same eigenspaces.
Now let $A$ and $B$ be simultaneously diagonalizable. Then there is a basis $(x_1,x_2)$ of vectors that are eigenvectors for both matrices. If one of the matrices is a multiple of the identity there is nothing to prove. If this is not the case then both matrices have two different eigenvalues with eigenspaces of dimension one. And the eigenspaces coincide.
For the $3times 3$ case, try to construct to matrices with a basis $(x_1,x_2,x_3)$ of common eigenvectors such that one matrix has two eigenspaces spanned by $x_1,x_2$ and $x_3$, the other has two eigenspaces spanned by $x_1$ and $x_2,x_3$.
answered Sep 5 at 14:16
daw
22.2k1542
add a comment |Â
up vote
0
down vote
accepted
up vote
0
down vote
accepted
If one of the matrices is a multiple of the identity then we have simultaneous diagonalizability. The same is true if they share the same eigenspaces.
Now let $A$ and $B$ be simultaneously diagonalizable. Then there is a basis $(x_1,x_2)$ of vectors that are eigenvectors for both matrices. If one of the matrices is a multiple of the identity there is nothing to prove. If this is not the case then both matrices have two different eigenvalues with eigenspaces of dimension one. And the eigenspaces coincide.
For the $3times 3$ case, try to construct to matrices with a basis $(x_1,x_2,x_3)$ of common eigenvectors such that one matrix has two eigenspaces spanned by $x_1,x_2$ and $x_3$, the other has two eigenspaces spanned by $x_1$ and $x_2,x_3$.
answered Sep 5 at 14:16
daw
22.2k1542
If one of the matrices is a multiple of the identity then we have simultaneous diagonalizability. The same is true if they share the same eigenspaces.
Now let $A$ and $B$ be simultaneously diagonalizable. Then there is a basis $(x_1,x_2)$ of vectors that are eigenvectors for both matrices. If one of the matrices is a multiple of the identity there is nothing to prove. If this is not the case then both matrices have two different eigenvalues with eigenspaces of dimension one. And the eigenspaces coincide.
For the $3times 3$ case, try to construct to matrices with a basis $(x_1,x_2,x_3)$ of common eigenvectors such that one matrix has two eigenspaces spanned by $x_1,x_2$ and $x_3$, the other has two eigenspaces spanned by $x_1$ and $x_2,x_3$.
answered Sep 5 at 14:16
daw
22.2k1542
answered Sep 5 at 14:16
daw
22.2k1542
answered Sep 5 at 14:16
daw
22.2k1542
answered Sep 5 at 14:16
daw
22.2k1542
22.2k1542
add a comment |Â
add a comment |Â
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