Mixing
Sum of positive semi-difinite matrix inequality
Clash Royale CLAN TAG#URR8PPP
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Can the following conclusion hold?
There exist matrices $H^iinmathbbR^mtimes n, (mleq n, iinmathcalN)$, and non-zero real numbers $underlineh^i$, and $overlineh^i$.
If the following inequality holds
beginalign
underlineh^i^2 I_mleq H^i (H^i)^T leq overlineh^i^2 I_m
endalign
and
beginalign
sum_iinmathcalN(H^i)^T H^i >0,
endalign
then
beginalign
sum_iinmathcalN(H^i)^T H^i geq min_iinmathcalNunderlineh^i^2 I_n
endalign
Note: The expression matrix $M>0$ means matrix $M$ is positive definite. Similarly, the expression $Ageq B$ means $A-B$ is positive semi-definite.
positive-semidefinite
asked Jul 20 at 12:11
wayne
359313
add a comment |Â
up vote
0
down vote
favorite
Can the following conclusion hold?
There exist matrices $H^iinmathbbR^mtimes n, (mleq n, iinmathcalN)$, and non-zero real numbers $underlineh^i$, and $overlineh^i$.
If the following inequality holds
beginalign
underlineh^i^2 I_mleq H^i (H^i)^T leq overlineh^i^2 I_m
endalign
and
beginalign
sum_iinmathcalN(H^i)^T H^i >0,
endalign
then
beginalign
sum_iinmathcalN(H^i)^T H^i geq min_iinmathcalNunderlineh^i^2 I_n
endalign
Note: The expression matrix $M>0$ means matrix $M$ is positive definite. Similarly, the expression $Ageq B$ means $A-B$ is positive semi-definite.
positive-semidefinite
asked Jul 20 at 12:11
wayne
359313
What does it mean for one matrix to be less than or equal to another matrix? Is $h$ a real number? Also, in the last line, what is the minimum being taken over?
– Chandler Watson
Jul 21 at 6:16
Thanks for your questions. The descriptions have been updated such that all your concerns have been addressed.
– wayne
Jul 21 at 12:00
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Can the following conclusion hold?
There exist matrices $H^iinmathbbR^mtimes n, (mleq n, iinmathcalN)$, and non-zero real numbers $underlineh^i$, and $overlineh^i$.
If the following inequality holds
beginalign
underlineh^i^2 I_mleq H^i (H^i)^T leq overlineh^i^2 I_m
endalign
and
beginalign
sum_iinmathcalN(H^i)^T H^i >0,
endalign
then
beginalign
sum_iinmathcalN(H^i)^T H^i geq min_iinmathcalNunderlineh^i^2 I_n
endalign
Note: The expression matrix $M>0$ means matrix $M$ is positive definite. Similarly, the expression $Ageq B$ means $A-B$ is positive semi-definite.
positive-semidefinite
asked Jul 20 at 12:11
wayne
359313
Can the following conclusion hold?
There exist matrices $H^iinmathbbR^mtimes n, (mleq n, iinmathcalN)$, and non-zero real numbers $underlineh^i$, and $overlineh^i$.
If the following inequality holds
beginalign
underlineh^i^2 I_mleq H^i (H^i)^T leq overlineh^i^2 I_m
endalign
and
beginalign
sum_iinmathcalN(H^i)^T H^i >0,
endalign
then
beginalign
sum_iinmathcalN(H^i)^T H^i geq min_iinmathcalNunderlineh^i^2 I_n
endalign
Note: The expression matrix $M>0$ means matrix $M$ is positive definite. Similarly, the expression $Ageq B$ means $A-B$ is positive semi-definite.
positive-semidefinite
asked Jul 20 at 12:11
wayne
359313
edited Jul 21 at 11:58
asked Jul 20 at 12:11
wayne
359313
asked Jul 20 at 12:11
wayne
359313
asked Jul 20 at 12:11
wayne
359313
359313
What does it mean for one matrix to be less than or equal to another matrix? Is $h$ a real number? Also, in the last line, what is the minimum being taken over?
– Chandler Watson
Jul 21 at 6:16
Thanks for your questions. The descriptions have been updated such that all your concerns have been addressed.
– wayne
Jul 21 at 12:00
add a comment |Â
What does it mean for one matrix to be less than or equal to another matrix? Is $h$ a real number? Also, in the last line, what is the minimum being taken over?
– Chandler Watson
Jul 21 at 6:16
Thanks for your questions. The descriptions have been updated such that all your concerns have been addressed.
– wayne
Jul 21 at 12:00
What does it mean for one matrix to be less than or equal to another matrix? Is $h$ a real number? Also, in the last line, what is the minimum being taken over?
– Chandler Watson
Jul 21 at 6:16
What does it mean for one matrix to be less than or equal to another matrix? Is $h$ a real number? Also, in the last line, what is the minimum being taken over?
– Chandler Watson
Jul 21 at 6:16
Thanks for your questions. The descriptions have been updated such that all your concerns have been addressed.
– wayne
Jul 21 at 12:00
Thanks for your questions. The descriptions have been updated such that all your concerns have been addressed.
– wayne
Jul 21 at 12:00
add a comment |Â
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What does it mean for one matrix to be less than or equal to another matrix? Is $h$ a real number? Also, in the last line, what is the minimum being taken over?
– Chandler Watson
Jul 21 at 6:16
Thanks for your questions. The descriptions have been updated such that all your concerns have been addressed.
– wayne
Jul 21 at 12:00