A determinantal lower bound Original Research Article

There are many known upper bounds for det(·), the modulus of the determinant function, but useful lower bounds are rare. We show that if ℓ and u are positive lower and upper bounds on the moduli of the eigenvalues of a real or complex invertible n×n matrix A such that tr(A)greater-or-equal, slantednℓ, then det(A)greater-or-equal, slantedℓκun−κ, where κ=κ(ℓ,u)=[(nu−tr(A))/(u−ℓ)]. Since κless-than-or-equals, slantn, our lower bound improves the obvious lower bound ℓn. In two cases the trace inequality is automatically satisfied. If A is a diagonally dominant real matrix with positive diagonal entries, then one can easily compute ℓ and u using Gerschgorinʹs circle theorem. While if A is positive definite Hermitian with eigenvalues λ1less-than-or-equals, slantλ2less-than-or-equals, slantcdots, three dots, centeredless-than-or-equals, slantλn, then our bound implies the inequality det(A)greater-or-equal, slantedλ1κλnn−κ, where κ=κ(λ1,λn). Lower bounds like those established here have proved useful in obtaining new formulas for the approximation of π, e, square roots of numbers, and more generally real or complex roots of arbitrary polynomials.