For real square matrices of the same dimension, does the following hold?
Clash Royale CLAN TAG#URR8PPP
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Given real square matrices $A_1,A_2$ of the same dimension, does the following inequality hold?
$e^A^T_1e^A^T_2e^A_2e^A_1leqe^A_1+A_2+A^T_1+A^T_2$
where $A^T_i$ stands for the transpose of $A_i$ and $leq$ denotes the negative semi-definiteness of the left argument with respect to right. I know if $A_1,A_2$ commute and both matrices are normal, then
$e^A^T_1e^A^T_2e^A_2e^A_1=e^A_1+A_2+A^T_1+A^T_2$. But I am not sure if the inequality mentioned above holds.
Any suggestions, hints or references relating to the results on similar matrix exponential inequalities are greatly appreciated.
linear-algebra matrices matrix-calculus matrix-decomposition matrix-exponential
asked Aug 30 at 0:26
jbgujgu
9210
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up vote
1
down vote
favorite
Given real square matrices $A_1,A_2$ of the same dimension, does the following inequality hold?
$e^A^T_1e^A^T_2e^A_2e^A_1leqe^A_1+A_2+A^T_1+A^T_2$
where $A^T_i$ stands for the transpose of $A_i$ and $leq$ denotes the negative semi-definiteness of the left argument with respect to right. I know if $A_1,A_2$ commute and both matrices are normal, then
$e^A^T_1e^A^T_2e^A_2e^A_1=e^A_1+A_2+A^T_1+A^T_2$. But I am not sure if the inequality mentioned above holds.
Any suggestions, hints or references relating to the results on similar matrix exponential inequalities are greatly appreciated.
linear-algebra matrices matrix-calculus matrix-decomposition matrix-exponential
asked Aug 30 at 0:26
jbgujgu
9210
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Given real square matrices $A_1,A_2$ of the same dimension, does the following inequality hold?
$e^A^T_1e^A^T_2e^A_2e^A_1leqe^A_1+A_2+A^T_1+A^T_2$
where $A^T_i$ stands for the transpose of $A_i$ and $leq$ denotes the negative semi-definiteness of the left argument with respect to right. I know if $A_1,A_2$ commute and both matrices are normal, then
$e^A^T_1e^A^T_2e^A_2e^A_1=e^A_1+A_2+A^T_1+A^T_2$. But I am not sure if the inequality mentioned above holds.
Any suggestions, hints or references relating to the results on similar matrix exponential inequalities are greatly appreciated.
linear-algebra matrices matrix-calculus matrix-decomposition matrix-exponential
asked Aug 30 at 0:26
jbgujgu
9210
Given real square matrices $A_1,A_2$ of the same dimension, does the following inequality hold?
$e^A^T_1e^A^T_2e^A_2e^A_1leqe^A_1+A_2+A^T_1+A^T_2$
where $A^T_i$ stands for the transpose of $A_i$ and $leq$ denotes the negative semi-definiteness of the left argument with respect to right. I know if $A_1,A_2$ commute and both matrices are normal, then
$e^A^T_1e^A^T_2e^A_2e^A_1=e^A_1+A_2+A^T_1+A^T_2$. But I am not sure if the inequality mentioned above holds.
Any suggestions, hints or references relating to the results on similar matrix exponential inequalities are greatly appreciated.
linear-algebra matrices matrix-calculus matrix-decomposition matrix-exponential
linear-algebra matrices matrix-calculus matrix-decomposition matrix-exponential
asked Aug 30 at 0:26
jbgujgu
9210
asked Aug 30 at 0:26
jbgujgu
9210
asked Aug 30 at 0:26
jbgujgu
9210
asked Aug 30 at 0:26
jbgujgu
9210
asked Aug 30 at 0:26
jbgujgu
9210
9210
add a comment |Â
add a comment |Â
1 Answer
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It's false. Take $A_1=0_2,A_2=beginpmatrix-7.2&-3.2\-0.2&-7.4endpmatrix$.
answered Sep 3 at 13:16
loup blanc
20.6k21649
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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's false. Take $A_1=0_2,A_2=beginpmatrix-7.2&-3.2\-0.2&-7.4endpmatrix$.
answered Sep 3 at 13:16
loup blanc
20.6k21649
add a comment |Â
up vote
1
down vote
accepted
It's false. Take $A_1=0_2,A_2=beginpmatrix-7.2&-3.2\-0.2&-7.4endpmatrix$.
answered Sep 3 at 13:16
loup blanc
20.6k21649
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
It's false. Take $A_1=0_2,A_2=beginpmatrix-7.2&-3.2\-0.2&-7.4endpmatrix$.
answered Sep 3 at 13:16
loup blanc
20.6k21649
It's false. Take $A_1=0_2,A_2=beginpmatrix-7.2&-3.2\-0.2&-7.4endpmatrix$.
answered Sep 3 at 13:16
loup blanc
20.6k21649
answered Sep 3 at 13:16
loup blanc
20.6k21649
answered Sep 3 at 13:16
loup blanc
20.6k21649
answered Sep 3 at 13:16
loup blanc
20.6k21649
20.6k21649
add a comment |Â
add a comment |Â
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