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Eigenvalues of $A +A^-1$
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Knowing the eigenvalues of $A$, what we can say about the eigenvalues of $A+A^-1$?
Is it true that $lambda_i,(A+A^-1) = lambda_i,A + frac1lambda_i,A$, where $lambda_i,A$ is the $i$-th eigenvalue of $A$?
I ask this question after asking this question, due to the different opinion in the conversation in the comment.
linear-algebra matrices eigenvalues-eigenvectors inverse
asked Jul 31 at 10:10
Alex
1729
add a comment |Â
up vote
0
down vote
favorite
Knowing the eigenvalues of $A$, what we can say about the eigenvalues of $A+A^-1$?
Is it true that $lambda_i,(A+A^-1) = lambda_i,A + frac1lambda_i,A$, where $lambda_i,A$ is the $i$-th eigenvalue of $A$?
I ask this question after asking this question, due to the different opinion in the conversation in the comment.
linear-algebra matrices eigenvalues-eigenvectors inverse
asked Jul 31 at 10:10
Alex
1729
Consider $A$ already in Jordan normal form. Without loss, consider it is reduced to one block. What can be said about $A+A^-1$ in this situation? (Please always show the own work!)
– dan_fulea
Jul 31 at 10:13
Yes sorry, i just asked this question due to the comments in the conversation in a my previous question!
– Alex
Jul 31 at 10:16
1
The other question was involving the transpose, there is no connection. In this case, please work out some simple examples, show at least why do you expect the claim to be true or false (mathematically). For instance for a diagonal matrix, then for a $2times 2$ matrix of the shape $$A = beginbmatrixa&1\0&aendbmatrix .$$
– dan_fulea
Jul 31 at 10:28
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Knowing the eigenvalues of $A$, what we can say about the eigenvalues of $A+A^-1$?
Is it true that $lambda_i,(A+A^-1) = lambda_i,A + frac1lambda_i,A$, where $lambda_i,A$ is the $i$-th eigenvalue of $A$?
I ask this question after asking this question, due to the different opinion in the conversation in the comment.
linear-algebra matrices eigenvalues-eigenvectors inverse
asked Jul 31 at 10:10
Alex
1729
Knowing the eigenvalues of $A$, what we can say about the eigenvalues of $A+A^-1$?
Is it true that $lambda_i,(A+A^-1) = lambda_i,A + frac1lambda_i,A$, where $lambda_i,A$ is the $i$-th eigenvalue of $A$?
I ask this question after asking this question, due to the different opinion in the conversation in the comment.
linear-algebra matrices eigenvalues-eigenvectors inverse
asked Jul 31 at 10:10
Alex
1729
edited Jul 31 at 10:12
asked Jul 31 at 10:10
Alex
1729
asked Jul 31 at 10:10
Alex
1729
asked Jul 31 at 10:10
Alex
1729
1729
Consider $A$ already in Jordan normal form. Without loss, consider it is reduced to one block. What can be said about $A+A^-1$ in this situation? (Please always show the own work!)
– dan_fulea
Jul 31 at 10:13
Yes sorry, i just asked this question due to the comments in the conversation in a my previous question!
– Alex
Jul 31 at 10:16
1
The other question was involving the transpose, there is no connection. In this case, please work out some simple examples, show at least why do you expect the claim to be true or false (mathematically). For instance for a diagonal matrix, then for a $2times 2$ matrix of the shape $$A = beginbmatrixa&1\0&aendbmatrix .$$
– dan_fulea
Jul 31 at 10:28
add a comment |Â
Consider $A$ already in Jordan normal form. Without loss, consider it is reduced to one block. What can be said about $A+A^-1$ in this situation? (Please always show the own work!)
– dan_fulea
Jul 31 at 10:13
Yes sorry, i just asked this question due to the comments in the conversation in a my previous question!
– Alex
Jul 31 at 10:16
1
The other question was involving the transpose, there is no connection. In this case, please work out some simple examples, show at least why do you expect the claim to be true or false (mathematically). For instance for a diagonal matrix, then for a $2times 2$ matrix of the shape $$A = beginbmatrixa&1\0&aendbmatrix .$$
– dan_fulea
Jul 31 at 10:28
Consider $A$ already in Jordan normal form. Without loss, consider it is reduced to one block. What can be said about $A+A^-1$ in this situation? (Please always show the own work!)
– dan_fulea
Jul 31 at 10:13
Consider $A$ already in Jordan normal form. Without loss, consider it is reduced to one block. What can be said about $A+A^-1$ in this situation? (Please always show the own work!)
– dan_fulea
Jul 31 at 10:13
Yes sorry, i just asked this question due to the comments in the conversation in a my previous question!
– Alex
Jul 31 at 10:16
Yes sorry, i just asked this question due to the comments in the conversation in a my previous question!
– Alex
Jul 31 at 10:16
1
1
The other question was involving the transpose, there is no connection. In this case, please work out some simple examples, show at least why do you expect the claim to be true or false (mathematically). For instance for a diagonal matrix, then for a $2times 2$ matrix of the shape $$A = beginbmatrixa&1\0&aendbmatrix .$$
– dan_fulea
Jul 31 at 10:28
The other question was involving the transpose, there is no connection. In this case, please work out some simple examples, show at least why do you expect the claim to be true or false (mathematically). For instance for a diagonal matrix, then for a $2times 2$ matrix of the shape $$A = beginbmatrixa&1\0&aendbmatrix .$$
– dan_fulea
Jul 31 at 10:28
add a comment |Â
1 Answer
1
active
oldest
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up vote
2
down vote
accepted
Consider a Schur decomposition of $A$: $A = UTU^*$, where $T$ is upper triangular with eigenvalues $lambda_j(A)$ on the diagonal and $U$ is unitary matrix. Clear that $A^-1 = U T^-1 U^*$, where $T^-1$ is again an upper triangular matrix with eigenvalues $1/lambda_j(A)$ of $A^-1$ on the diagonal. Then $A + A^-1 = U (T + T^-1) U^*$ which implies that eigenvalues of $A + A^-1$ are exactly $lambda_j(A) + 1/lambda_j(A)$
answered Jul 31 at 10:26
Stanislav Morozov
6721417
Why do you not allow Alex to make his own effort to elucidate the structure? The question comes with no context. No effort / example is shown in the OP. It is maybe not a good idea to give a full answer. By the way, the OP does not say anything about the field of definition of the matrix $A$. (Why it should be defined over a field of characteristic zero includded in $Bbb C$?)
– dan_fulea
Jul 31 at 10:35
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Consider a Schur decomposition of $A$: $A = UTU^*$, where $T$ is upper triangular with eigenvalues $lambda_j(A)$ on the diagonal and $U$ is unitary matrix. Clear that $A^-1 = U T^-1 U^*$, where $T^-1$ is again an upper triangular matrix with eigenvalues $1/lambda_j(A)$ of $A^-1$ on the diagonal. Then $A + A^-1 = U (T + T^-1) U^*$ which implies that eigenvalues of $A + A^-1$ are exactly $lambda_j(A) + 1/lambda_j(A)$
answered Jul 31 at 10:26
Stanislav Morozov
6721417
Why do you not allow Alex to make his own effort to elucidate the structure? The question comes with no context. No effort / example is shown in the OP. It is maybe not a good idea to give a full answer. By the way, the OP does not say anything about the field of definition of the matrix $A$. (Why it should be defined over a field of characteristic zero includded in $Bbb C$?)
– dan_fulea
Jul 31 at 10:35
add a comment |Â
up vote
2
down vote
accepted
Consider a Schur decomposition of $A$: $A = UTU^*$, where $T$ is upper triangular with eigenvalues $lambda_j(A)$ on the diagonal and $U$ is unitary matrix. Clear that $A^-1 = U T^-1 U^*$, where $T^-1$ is again an upper triangular matrix with eigenvalues $1/lambda_j(A)$ of $A^-1$ on the diagonal. Then $A + A^-1 = U (T + T^-1) U^*$ which implies that eigenvalues of $A + A^-1$ are exactly $lambda_j(A) + 1/lambda_j(A)$
answered Jul 31 at 10:26
Stanislav Morozov
6721417
Why do you not allow Alex to make his own effort to elucidate the structure? The question comes with no context. No effort / example is shown in the OP. It is maybe not a good idea to give a full answer. By the way, the OP does not say anything about the field of definition of the matrix $A$. (Why it should be defined over a field of characteristic zero includded in $Bbb C$?)
– dan_fulea
Jul 31 at 10:35
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Consider a Schur decomposition of $A$: $A = UTU^*$, where $T$ is upper triangular with eigenvalues $lambda_j(A)$ on the diagonal and $U$ is unitary matrix. Clear that $A^-1 = U T^-1 U^*$, where $T^-1$ is again an upper triangular matrix with eigenvalues $1/lambda_j(A)$ of $A^-1$ on the diagonal. Then $A + A^-1 = U (T + T^-1) U^*$ which implies that eigenvalues of $A + A^-1$ are exactly $lambda_j(A) + 1/lambda_j(A)$
answered Jul 31 at 10:26
Stanislav Morozov
6721417
Consider a Schur decomposition of $A$: $A = UTU^*$, where $T$ is upper triangular with eigenvalues $lambda_j(A)$ on the diagonal and $U$ is unitary matrix. Clear that $A^-1 = U T^-1 U^*$, where $T^-1$ is again an upper triangular matrix with eigenvalues $1/lambda_j(A)$ of $A^-1$ on the diagonal. Then $A + A^-1 = U (T + T^-1) U^*$ which implies that eigenvalues of $A + A^-1$ are exactly $lambda_j(A) + 1/lambda_j(A)$
answered Jul 31 at 10:26
Stanislav Morozov
6721417
answered Jul 31 at 10:26
Stanislav Morozov
6721417
answered Jul 31 at 10:26
Stanislav Morozov
6721417
answered Jul 31 at 10:26
Stanislav Morozov
6721417
6721417
Why do you not allow Alex to make his own effort to elucidate the structure? The question comes with no context. No effort / example is shown in the OP. It is maybe not a good idea to give a full answer. By the way, the OP does not say anything about the field of definition of the matrix $A$. (Why it should be defined over a field of characteristic zero includded in $Bbb C$?)
– dan_fulea
Jul 31 at 10:35
add a comment |Â
Why do you not allow Alex to make his own effort to elucidate the structure? The question comes with no context. No effort / example is shown in the OP. It is maybe not a good idea to give a full answer. By the way, the OP does not say anything about the field of definition of the matrix $A$. (Why it should be defined over a field of characteristic zero includded in $Bbb C$?)
– dan_fulea
Jul 31 at 10:35
Why do you not allow Alex to make his own effort to elucidate the structure? The question comes with no context. No effort / example is shown in the OP. It is maybe not a good idea to give a full answer. By the way, the OP does not say anything about the field of definition of the matrix $A$. (Why it should be defined over a field of characteristic zero includded in $Bbb C$?)
– dan_fulea
Jul 31 at 10:35
Why do you not allow Alex to make his own effort to elucidate the structure? The question comes with no context. No effort / example is shown in the OP. It is maybe not a good idea to give a full answer. By the way, the OP does not say anything about the field of definition of the matrix $A$. (Why it should be defined over a field of characteristic zero includded in $Bbb C$?)
– dan_fulea
Jul 31 at 10:35
add a comment |Â
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Consider $A$ already in Jordan normal form. Without loss, consider it is reduced to one block. What can be said about $A+A^-1$ in this situation? (Please always show the own work!)
– dan_fulea
Jul 31 at 10:13
Yes sorry, i just asked this question due to the comments in the conversation in a my previous question!
– Alex
Jul 31 at 10:16
1
The other question was involving the transpose, there is no connection. In this case, please work out some simple examples, show at least why do you expect the claim to be true or false (mathematically). For instance for a diagonal matrix, then for a $2times 2$ matrix of the shape $$A = beginbmatrixa&1\0&aendbmatrix .$$
– dan_fulea
Jul 31 at 10:28