# Inverse Hadamard determinant inequality

As far as I remembered there is an inverse Hadamard inequality for the determinant of the form $$|D|>\prod_j \sqrt{(a_{jj}^2-\sum_{i\neq j}a_{ij}^2)}$$ providing all values in $(\cdot)>0$.

Please help me with exact references to this inequality, its possible generalizations and modifications and comments.

## 1 Answer

Did you mean Ostrowski's theorem $$|D|>\prod_j \left(|a_{jj}|-\sum_{i\neq j}|a_{ij}|\right)$$ for diagonally dominant matrices (see e.g. http://planetmath.org/propertiesofdiagonallydominantmatrix)?

• Yes, it is exactly what I wanted, user35593. Thank you very much! – Sergei Apr 6 '16 at 18:14
• Do you know a reference to Ostrowski's paper? Is not a first proposed inequality a consequence? – Sergei Apr 6 '16 at 18:29
• Perhaps it is Ostrowski, Alexander; Über die determinanten mit überwiegender Hauptdiagonale. Comment. Math. Helv. 10 (1937), no. 1, 69–96, or A. M. Ostrowski, "Sur la determination des borns inferieures pour une class des determinants," Bulletin des Sciences Mathmatiques, vol. 61, no. 2, pp. 19–32, 1937. – Gerry Myerson Apr 6 '16 at 23:31
• @Gerry Myerson - thank you. May somebody help with a paper in Bulletin des Sciences Mathematiques? A first paper-not an easy reading... – Sergei Apr 7 '16 at 7:32
• Is the next generalization valid with two pivoted elements: let $(a_1,a_2,a_3,\dots)$ be a line of a determinant with $a_1+a_2>\sum_{k=3}^n a_k$. Is the Ostrowski inequality still valid with a product of \prod(a_1+a_2-\sum_{k=3}^n a_k)\$ -??? – Sergei Apr 18 '16 at 16:46