If 0 <= X <= Id and 0<=A, then XAX <= A?

Elliana Molina

Elliana Molina

Answered question

2022-11-13

Let X , A C n × n and suppose
0 X Id , 0 A ,
where Id C n × n denotes the identity matrix. Is it true that
X A X A ,
or can you give a counterexample?

Answer & Explanation

Stella Andrade

Stella Andrade

Beginner2022-11-14Added 19 answers

Here seems to be a counterexample. Let
A = ( 1 1 1 1 ) ,   X = ( 0 0 0 1 ) .
We have 0 A, 0 X I d, X A X = X, and
B = A X A X = A X = ( 1 1 1 0 ) .
Matrix B is not positive definite matrix.

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