1
$\begingroup$

$A=[a_{ij}]\in M_n(\mathbb{R})$ and $c=\max\{|a_{ij}|\}$

I need to show $|\det A|\le c^nn^{n/2}$

Let $\lambda_1,\dots,\lambda_n$ be the eigenvalues of $A$

I can say that $|\det A|^{2/n}=|\lambda_1\cdots\lambda_n|^{2/n}\le{|\lambda_1|^2+\dots+|\lambda_n|^2\over n} $

could anyone tell me how to proceed next?

$\endgroup$
  • $\begingroup$ See Hadamard's inequality. $\endgroup$ – Julien Nov 3 '13 at 15:08
  • $\begingroup$ What If I just want to prove myself from the given data? Thank you though :) $\endgroup$ – Marso Nov 3 '13 at 15:10
4
$\begingroup$

The absolute value of a determinant expresses the volume of the parallelepiped spanned by its columns. In this case each column has length no more than $c \sqrt{n}$ and therefore the volume is no more than $(c\sqrt{n})^n$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.