Theoretische Chemie I

Theoretische Chemie I

Wichtige Grundlagen der linearen Algebra

Ubungen (Teil 3) ¨

Mathematik ist das Alphabet, mit dessen Hilfe Gott das Universum beschrieben hat Galileo Galilei

Aufgabe 1

Verify the following relations for matrix exponentials:

a)

exp(A)^†= exp(A^†) b)

Bexp(A)B⁻¹ = exp(BAB⁻¹) c)

d

dλexp(λA) = Aexp(λA) = exp(λA)A provided the exponential of a matrix A is defined as

exp(A) =

∞

X

n=0

Aⁿ n!

Aufgabe 2

LetD be a diagonal matrix with diagonal elementsd_i:

D= diag(d₁, d₂, ..., d_n) (1)











d₁ 0 · · · 0

0 d₂ · · · 0

... . .. ...

0 0 · · · d_n











(2)

(a) Show that the exponential of the diagonal matrix is given by:

exp(D) = diag(exp(d₁),exp(d₂), ...,exp(d_n)) (3) (b) Let A be a matrix that can be diagonalized:

A=XDX⁻¹ (4)

Verify the following relation for the determinant of the exponential of A:

det[exp(A)] = exp(TrA) (5)

where TrA is the trace of the matrix A:

TrA=

n

X

i=1

Aii (6)