determinant in terms of traces of powers

It is possible to express the determinantMathworldPlanetmath of a matrix in of traces of powers of a matrix.

The easiest way to derive these expressions is to specialize to the case of diagonal matricesMathworldPlanetmath. For instance, suppose we have a 2×2 matrix M=diag(u,v). Then

detM = uv
trM = u+v
trM2 = u2+v2

From the algebraic identity (u+v)2=u2+v2+2uv, it can be concluded that detM=12(trM)2-12tr(M2).

Likewise, one can derive expressions for the determinants of larger matrices from the identities for elementary symmetric polynomials in of power sums. For instance, from the identity


it can be concluded that


for a 3×3 matrix M.

Title determinant in terms of traces of powers
Canonical name DeterminantInTermsOfTracesOfPowers
Date of creation 2013-03-22 15:57:08
Last modified on 2013-03-22 15:57:08
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 11
Author Mathprof (13753)
Entry type Theorem
Classification msc 15A15