Linear matrix inequality given the bounds If A\succeq A_{\min}\succ 0, B\succeq

acceloq3c

acceloq3c

Answered question

2022-04-30

Linear matrix inequality given the bounds
If AAmin0,BBmin0, will the following be true?
ABAABminA
How can I prove it?

Answer & Explanation

elvis0217t2x

elvis0217t2x

Beginner2022-05-01Added 13 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 A0, for any vector xRN and x0, we have Ax0 because A has full rank. Thus xA(B-Bmin)Ax>0. Therefore A(B-Bmin)A0 and ABAABminA.

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