Inequalities 4)-6) are known as ”Hadamard’s inequalities”.
(Note that inequalities 2)-5) may suggest the idea that such inequalities could hold: or for any ; however, this is not true, as one can easily see with and . Actually, inequalities 2)-5) give the best possible estimate of this kind.)
2) If is singular, the thesis is trivial. Let then . Let’s define , ,. (Note that exist for any , because implies no all-zero row exists.) So and, since , we have:
3) Same as 2), but applied to .
4)-6) See related proofs attached to ”Hadamard’s inequalities”.
|Date of creation||2013-03-22 15:34:46|
|Last modified on||2013-03-22 15:34:46|
|Owner||Andrea Ambrosio (7332)|
|Last modified by||Andrea Ambrosio (7332)|
|Author||Andrea Ambrosio (7332)|