determinant inequalities

There are a number of interesting inequalities bounding the determinantMathworldPlanetmath of a n×n complex matrix A, where ρ is its spectral radius:

1) |det(A)|ρn(A)
2) |det(A)|i=1n(j=1n|aij|)=i=1nai1
3) |det(A)|j=1n(i=1n|aij|)=j=1naj1
4) |det(A)|i=1n(j=1n|aij|2)12=i=1nai2
5) |det(A)|j=1n(i=1n|aij|2)12=j=1naj2
6) if A is Hermitian positive semidefinitePlanetmathPlanetmath, det(A)i=1naii, with equality if and only if A is diagonal.

Inequalities 4)-6) are known as ”Hadamard’s inequalities”.

(Note that inequalities 2)-5) may suggest the idea that such inequalities could hold: |det(A)|i=1naip or |det(A)|j=1najp for any p𝐍; however, this is not true, as one can easily see with A=[11-11] and p=3. Actually, inequalities 2)-5) give the best possible estimate of this kind.)


1) |det(A)|=|i=1nλi|=i=1n|λi|i=1nρ(A)=ρn(A).

2) If A is singular, the thesis is trivial. Let then det(A)0. Let’s define B=DA, D=diag(d11,d22,,dnn),dii=(j=1n|aij|)-1. (Note that dii exist for any i, because det(A)0 implies no all-zero row exists.) So B=maxi(j=1n|bij|)=1 and, since ρ(B)B, we have:


from which:


3) Same as 2), but applied to AT.

4)-6) See related proofs attached to ”Hadamard’s inequalities”.

Title determinant inequalities
Canonical name DeterminantInequalities
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)
Numerical id 12
Author Andrea Ambrosio (7332)
Entry type Result
Classification msc 15A15