Linear matrix inequality given the bounds If \(\displaystyle{A}\succ{e}{q}{A}_{{\min}}\succ{0},{B}\succ{e}{q}{B}_{{\min}}\succ{0}\),

Kiara Haas

Kiara Haas

Answered question

2022-03-21

Linear matrix inequality given the bounds
If AeqAmin0,BeqBmin0, will the following be true?
ABAeqABminA
How can I prove it?

Answer & Explanation

membatas0v2v

membatas0v2v

Beginner2022-03-22Added 19 answers

We can prove by using definition of positive-definiteness. For any vector vRN and v0, it's obvious that from the definition v(BBmin)v0. Since Aeq0, for any vector xRN and x0, we have Ax0 because A has full rank. Thus xA(BBmin)Ax>0. Therefore A(BBmin)Aeq0 and ABAeqABminA.

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