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How to determine two matrices are conjugate.
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0
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Which of the following statements are true?
- The matrices $
A=left[ beginarraycc
1 & 1 \
0 & 1\
endarray right]$ and $
B=left[ beginarraycc
1 & 0 \
1 & 1\
endarray right]$ are conjugate in $GL_2(mathbbR)$ - The matrices $
A=left[ beginarraycc
1 & 1 \
0 & 1\
endarray right]$ and $
B=left[ beginarraycc
1 & 0 \
1 & 1\
endarray right]$ are conjugate in $SL_2(mathbbR)$ - The matrices $
C=left[ beginarraycc
1 & 0 \
0 & 2\
endarray right]$ and $
D=left[ beginarraycc
1 & 3 \
0 & 2\
endarray right]$ are conjugate in $GL_2(mathbbR)$
I know the conjugate matrices have the same eigenvalues. But all of this matrices have the same eigenvalues. I am not sure how to determine that which matrices are conjugate to each other.
linear-algebra matrices eigenvalues-eigenvectors
asked Jul 23 at 8:44
Babai
2,50021539
add a comment |Â
up vote
0
down vote
favorite
Which of the following statements are true?
- The matrices $
A=left[ beginarraycc
1 & 1 \
0 & 1\
endarray right]$ and $
B=left[ beginarraycc
1 & 0 \
1 & 1\
endarray right]$ are conjugate in $GL_2(mathbbR)$ - The matrices $
A=left[ beginarraycc
1 & 1 \
0 & 1\
endarray right]$ and $
B=left[ beginarraycc
1 & 0 \
1 & 1\
endarray right]$ are conjugate in $SL_2(mathbbR)$ - The matrices $
C=left[ beginarraycc
1 & 0 \
0 & 2\
endarray right]$ and $
D=left[ beginarraycc
1 & 3 \
0 & 2\
endarray right]$ are conjugate in $GL_2(mathbbR)$
I know the conjugate matrices have the same eigenvalues. But all of this matrices have the same eigenvalues. I am not sure how to determine that which matrices are conjugate to each other.
linear-algebra matrices eigenvalues-eigenvectors
asked Jul 23 at 8:44
Babai
2,50021539
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Which of the following statements are true?
- The matrices $
A=left[ beginarraycc
1 & 1 \
0 & 1\
endarray right]$ and $
B=left[ beginarraycc
1 & 0 \
1 & 1\
endarray right]$ are conjugate in $GL_2(mathbbR)$ - The matrices $
A=left[ beginarraycc
1 & 1 \
0 & 1\
endarray right]$ and $
B=left[ beginarraycc
1 & 0 \
1 & 1\
endarray right]$ are conjugate in $SL_2(mathbbR)$ - The matrices $
C=left[ beginarraycc
1 & 0 \
0 & 2\
endarray right]$ and $
D=left[ beginarraycc
1 & 3 \
0 & 2\
endarray right]$ are conjugate in $GL_2(mathbbR)$
I know the conjugate matrices have the same eigenvalues. But all of this matrices have the same eigenvalues. I am not sure how to determine that which matrices are conjugate to each other.
linear-algebra matrices eigenvalues-eigenvectors
asked Jul 23 at 8:44
Babai
2,50021539
Which of the following statements are true?
- The matrices $
A=left[ beginarraycc
1 & 1 \
0 & 1\
endarray right]$ and $
B=left[ beginarraycc
1 & 0 \
1 & 1\
endarray right]$ are conjugate in $GL_2(mathbbR)$ - The matrices $
A=left[ beginarraycc
1 & 1 \
0 & 1\
endarray right]$ and $
B=left[ beginarraycc
1 & 0 \
1 & 1\
endarray right]$ are conjugate in $SL_2(mathbbR)$ - The matrices $
C=left[ beginarraycc
1 & 0 \
0 & 2\
endarray right]$ and $
D=left[ beginarraycc
1 & 3 \
0 & 2\
endarray right]$ are conjugate in $GL_2(mathbbR)$
I know the conjugate matrices have the same eigenvalues. But all of this matrices have the same eigenvalues. I am not sure how to determine that which matrices are conjugate to each other.
linear-algebra matrices eigenvalues-eigenvectors
asked Jul 23 at 8:44
Babai
2,50021539
asked Jul 23 at 8:44
Babai
2,50021539
asked Jul 23 at 8:44
Babai
2,50021539
asked Jul 23 at 8:44
Babai
2,50021539
2,50021539
add a comment |Â
add a comment |Â
1 Answer
1
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0
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Hint
The definition of conjugacy is: if $A$ and $B$ are two matrices such that $$A= P^-1BP$$ then they are conjugate. In $GL(2,mathbbR)$ is the same as similarity so not only the eigenvalues have to be preserved: mainly two similar matrices in $GL(2,mathbbR)$ have the same Jordan normal form
Notice that A is in Jordan canonical form (just look at the eigenvalues). Notice that the $D$ matrix is diagonalisable
Probabily this forum Why aren't this two matrices conjugate in $SL(2,mathbbR)$ can help for the second question.
answered Jul 23 at 9:09
Davide Morgante
1,775220
What is $D$ here?
– Babai
Jul 23 at 9:15
Sorry, I'm indicating the matrices as they are on your worksheet $$ A=left[beginmatrix 1&1\0&1endmatrixright]$$ and $$ D=left[beginmatrix 1&3\0&2endmatrixright]$$
– Davide Morgante
Jul 23 at 9:17
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Hint
The definition of conjugacy is: if $A$ and $B$ are two matrices such that $$A= P^-1BP$$ then they are conjugate. In $GL(2,mathbbR)$ is the same as similarity so not only the eigenvalues have to be preserved: mainly two similar matrices in $GL(2,mathbbR)$ have the same Jordan normal form
Notice that A is in Jordan canonical form (just look at the eigenvalues). Notice that the $D$ matrix is diagonalisable
Probabily this forum Why aren't this two matrices conjugate in $SL(2,mathbbR)$ can help for the second question.
answered Jul 23 at 9:09
Davide Morgante
1,775220
What is $D$ here?
– Babai
Jul 23 at 9:15
Sorry, I'm indicating the matrices as they are on your worksheet $$ A=left[beginmatrix 1&1\0&1endmatrixright]$$ and $$ D=left[beginmatrix 1&3\0&2endmatrixright]$$
– Davide Morgante
Jul 23 at 9:17
add a comment |Â
up vote
0
down vote
Hint
The definition of conjugacy is: if $A$ and $B$ are two matrices such that $$A= P^-1BP$$ then they are conjugate. In $GL(2,mathbbR)$ is the same as similarity so not only the eigenvalues have to be preserved: mainly two similar matrices in $GL(2,mathbbR)$ have the same Jordan normal form
Notice that A is in Jordan canonical form (just look at the eigenvalues). Notice that the $D$ matrix is diagonalisable
Probabily this forum Why aren't this two matrices conjugate in $SL(2,mathbbR)$ can help for the second question.
answered Jul 23 at 9:09
Davide Morgante
1,775220
What is $D$ here?
– Babai
Jul 23 at 9:15
Sorry, I'm indicating the matrices as they are on your worksheet $$ A=left[beginmatrix 1&1\0&1endmatrixright]$$ and $$ D=left[beginmatrix 1&3\0&2endmatrixright]$$
– Davide Morgante
Jul 23 at 9:17
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Hint
The definition of conjugacy is: if $A$ and $B$ are two matrices such that $$A= P^-1BP$$ then they are conjugate. In $GL(2,mathbbR)$ is the same as similarity so not only the eigenvalues have to be preserved: mainly two similar matrices in $GL(2,mathbbR)$ have the same Jordan normal form
Notice that A is in Jordan canonical form (just look at the eigenvalues). Notice that the $D$ matrix is diagonalisable
Probabily this forum Why aren't this two matrices conjugate in $SL(2,mathbbR)$ can help for the second question.
answered Jul 23 at 9:09
Davide Morgante
1,775220
Hint
The definition of conjugacy is: if $A$ and $B$ are two matrices such that $$A= P^-1BP$$ then they are conjugate. In $GL(2,mathbbR)$ is the same as similarity so not only the eigenvalues have to be preserved: mainly two similar matrices in $GL(2,mathbbR)$ have the same Jordan normal form
Notice that A is in Jordan canonical form (just look at the eigenvalues). Notice that the $D$ matrix is diagonalisable
Probabily this forum Why aren't this two matrices conjugate in $SL(2,mathbbR)$ can help for the second question.
answered Jul 23 at 9:09
Davide Morgante
1,775220
answered Jul 23 at 9:09
Davide Morgante
1,775220
answered Jul 23 at 9:09
Davide Morgante
1,775220
answered Jul 23 at 9:09
Davide Morgante
1,775220
1,775220
What is $D$ here?
– Babai
Jul 23 at 9:15
Sorry, I'm indicating the matrices as they are on your worksheet $$ A=left[beginmatrix 1&1\0&1endmatrixright]$$ and $$ D=left[beginmatrix 1&3\0&2endmatrixright]$$
– Davide Morgante
Jul 23 at 9:17
add a comment |Â
What is $D$ here?
– Babai
Jul 23 at 9:15
Sorry, I'm indicating the matrices as they are on your worksheet $$ A=left[beginmatrix 1&1\0&1endmatrixright]$$ and $$ D=left[beginmatrix 1&3\0&2endmatrixright]$$
– Davide Morgante
Jul 23 at 9:17
What is $D$ here?
– Babai
Jul 23 at 9:15
What is $D$ here?
– Babai
Jul 23 at 9:15
Sorry, I'm indicating the matrices as they are on your worksheet $$ A=left[beginmatrix 1&1\0&1endmatrixright]$$ and $$ D=left[beginmatrix 1&3\0&2endmatrixright]$$
– Davide Morgante
Jul 23 at 9:17
Sorry, I'm indicating the matrices as they are on your worksheet $$ A=left[beginmatrix 1&1\0&1endmatrixright]$$ and $$ D=left[beginmatrix 1&3\0&2endmatrixright]$$
– Davide Morgante
Jul 23 at 9:17
add a comment |Â
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