556 Chapter 19The matrix eigenvalue problem
20.The Hückel Hamiltonian matrix of butadiene is
Find (i)the eigenvalues (in terms of αand β), (ii)the orthonormal eigenvectors.
You may find the following relations useful: (the ‘golden section’,
see Section 7.2), ,,φ
2
1 − 111 = 1 φ,φ
2
1 − 1 φ 1 = 1 1.
21.The Hückel Hamiltonian matrix of cyclopropene is
(i) Show that the eigenvalues areE
1
1 = 1 α 1 + 12 β,E
2
1 = 1 E
3
1 = 1 α 1 − 1 β. (ii) Find the normalized
eigenvector belonging to eigenvalueE
1
. (iii) Show that an eigenvector belonging to the
doubly-degenerate eigenvalueα 1 − 1 βhas componentsx
1
, x
2
, x
3
that satisfyx
1
1 + 1 x
2
1 + 1 x
3
1 = 10.
(iv) Find two orthonormal eigenvectors corresponding to eigenvalueα 1 − 1 β(you may
find the results of Examples 19.4(ii) and 19.8 useful).
- (i)Show that a square matrix Aand its transposeA
T
have the same set of eigenvalues.
(ii) Show that the following two equations are equivalent:
A
T
y 1 = 1 λy, y
T
A 1 = 1 λy
T
The eigenvectors ofA
T
are in general different from those of A(unless Ais symmetric).
The vector y is sometimes called a left-eigenvectorof A, and an ‘ordinary’ eigenvector x
of Ais then called a right-eigenvector.
(iii)Find the eigenvalues and corresponding normalized right- and left-eigenvectors of
.
Section 19.4
23.For the matrix
of Exercise 8, construct (i) the matrix Xof the eigenvectors and (ii) the diagonal matrixD
of the eigenvalues.(iii) Show thatAX 1 = 1 DX.
24.Repeat Exercise 23 for the matrix of Exercise 9.
25.For the matrix of Exercise 7,
(i)construct the matrix Xof the eigenfunctions ofA, and find its inverse,X
− 1
,
(ii)calculateD 1 = 1 X
− 1
AXand confirm that Dis the diagonal matrix of the eigenvalues
ofA.
A=
31
13
030
303
030
A=
120
210
021
A=
32
02
αββ
βα β
ββα
φφ−=15121()− =
φ2
=+() 532
φ=+()512
αβ
βα β
βα β
βα
00
0
0
00