Sum of positive semi-difinite matrix inequality

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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




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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














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




share|cite|improve this question





















  • 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












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




share|cite|improve this question













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





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 21 at 11:58
























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
















  • 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















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