Let L:(−1/2,1/2) -> M(n xx n) be a continuous matrix-valued function such that each L(t) is symmetric and positive definite. Define M_k=int_(-1/2)^(1/2) t^k L(t)dt I want to prove or disprove that M_2 >= M_1M_0^(−1) M_1

drogaid1d8

drogaid1d8

Answered question

2022-11-04

Let L : ( 1 2 , 1 2 ) M ( n × n ) be a continuous matrix-valued function such that each L(t) is symmetric and positive definite.
Define
M k = 1 2 1 2 t k L ( t ) d t
I want to prove or disprove that
M 2 M 1 M 0 1 M 1
In other words, for all x R n we have
M 2 x , x M 1 M 0 1 M 1 x , x
This is obviously true for n=1 since by Cauchy-Schwarz inequality we have
( 1 2 1 2 t L ( t ) d t ) 2 ( 1 2 1 2 t 2 L ( t ) d t ) ( 1 2 1 2 L ( t ) d t )
That is M 1 2 M 2 M 0 .

Answer & Explanation

Leo Robinson

Leo Robinson

Beginner2022-11-05Added 14 answers

First note that M 0 and M 2 are positive definite. By the Schur complement characterization of block positive semidefinite matrices your matrix inequality is equivalent to
Z = ( M 0 M 1 M 1 M 2 ) 0 .
So it suffices to show Z 0 and we have
Z = ( 1 / 2 1 / 2 L ( t ) d t 1 / 2 1 / 2 t L ( t ) d t 1 / 2 1 / 2 t L ( t ) d t 1 / 2 1 / 2 t 2 L ( t ) d t ) = 1 / 2 1 / 2 ( L ( t ) t L ( t ) t L ( t ) t 2 L ( t ) ) d t = 1 / 2 1 / 2 ( I t I ) L ( t ) ( I t I ) d t = 1 / 2 1 / 2 N ( t ) L ( t ) N ( t ) T d t
where on the final line we defined N ( t ) = ( I t I ) . Note that as L(t)>0 for all t we have N ( t ) L ( t ) N ( t ) T 0 for all t and thus Z 0 and we are done.

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