proof that deteA=etrA

According to Schur decompositionMathworldPlanetmath the matrix A can be written after a suitable change of basis as A=D+N where D is a diagonal matrixMathworldPlanetmath and N is a strictly upper triangular matrixMathworldPlanetmath.

The formula we aim to prove


is invariant under a change of basis and thus we can carry out the computation of the exponentialPlanetmathPlanetmath in any basis we choose.

By definition

eA=n=0Ann! (1)

By the properties of diagonal and strictly upper triangular matrices we know that both DN and ND will also be strictly upper triangular matrices and so will their sum.

Thus the powers of A are of the form:

A = (D+N)=D+N1 (2)
A2 = (D+N)(D+N)=D2+N2 (3)
A3 = (D+N)(D2+N2)=D3+N3 (4)
Ak = Dk+Nk (6)

where all the Ni matrices are strictly upper triangular. Explicitly, N2=DN1+N1D+N12 and by recursion Nn+1=DNn+NnD+N1Nn.

Using equation 1 we can write

eA=eD+N~ (8)

where N~=n=1Nnn! is strictly upper triangular and eD=diag(eλ1,,eλn), where D=diag(λ1,,λn).

eA will thus be an upper triangular matrix. Since the determinantMathworldPlanetmath of an upper triangular matrix is just the product of the elements in its diagonal, we can write:

deteA=i=1neλi=ei=1nλi=etrA (9)
Title proof that deteA=etrA
Canonical name ProofThatdetEAEoperatornametrA
Date of creation 2013-03-22 15:51:56
Last modified on 2013-03-22 15:51:56
Owner cvalente (11260)
Last modified by cvalente (11260)
Numerical id 7
Author cvalente (11260)
Entry type Proof
Classification msc 15-00
Classification msc 15A15
Related topic SchurDecomposition