Is it true that for all symmetric n &#x00D7;<!-- × --> n matrix, ( a <

Marlie Cole

Marlie Cole

Answered question

2022-06-02

Is it true that for all symmetric n × n matrix, ( a i j ), such that for x { ± 1 } n
i j a i j x i x j 1
there exists a universal constant such that
i j a i j x i y j C
for all x , y { ± 1 } n . I tried to use the polarization identity
A x , y = A u , u A v , v
where u = ( x + y ) / 2 and v = ( x y ) / 2. However, as x and y vary over ± 1 vectors, u and v can be vectors in { ± 1 , 0 }.

Answer & Explanation

Jamir Rojas

Jamir Rojas

Beginner2022-06-03Added 1 answers

No, e.g. x T ( n n ) x 0 for every { 1 , 1 }-vector x, but ( 1 , 1 ) ( n n ) ( 1 1 ) = 2 n is unbounded.

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