8.19 EXERCISES
8.20 Demonstrate that the matrix
A=
200
− 644
3 − 10
is defective, i.e. does not have three linearly independent eigenvectors, by showing
the following:
(a) its eigenvalues are degenerate and, in fact, all equal;
(b) any eigenvector has the form (μ (3μ− 2 ν) ν)T.
(c) if two pairs of values,μ 1 ,ν 1 andμ 2 ,ν 2 , define two independent eigenvectors
v 1 andv 2 ,thenanythird similarly defined eigenvectorv 3 canbewrittenas
a linear combination ofv 1 andv 2 ,i.e.
v 3 =av 1 +bv 2 ,
where
a=μ 3 ν 2 −μ 2 ν 3
μ 1 ν 2 −μ 2 ν 1and b=μ 1 ν 3 −μ 3 ν 1
μ 1 ν 2 −μ 2 ν 1.
Illustrate (c) using the example (μ 1 ,ν 1 )=(1,1),(μ 2 ,ν 2 )=(1,2) and (μ 3 ,ν 3 )=
(0,1).
Show further that any matrix of the form
200
6 n− 64 − 2 n 4 − 4 n
3 − 3 nn− 12 n
is defective, with the same eigenvalues and eigenvectors asA.
8.21 By finding the eigenvectors of the Hermitian matrix
H=(
10 3 i
− 3 i 2)
,
construct a unitary matrixUsuch thatU†HU= Λ, where Λ is a real diagonal
matrix.
8.22 Use the stationary properties of quadratic forms to determine the maximum and
minimum values taken by the expression
Q=5x^2 +4y^2 +4z^2 +2xz+2xy
on the unit sphere,x^2 +y^2 +z^2 = 1. For what values ofx,yandzdo they occur?
8.23 Given that the matrix
A=
2 − 10
− 12 − 1
0 − 12
has two eigenvectors of the form (1 y 1)T, use the stationary property of the
expressionJ(x)=xTAx/(xTx) to obtain the corresponding eigenvalues. Deduce
the third eigenvalue.
8.24 Find the lengths of the semi-axes of the ellipse
73 x^2 +72xy+52y^2 = 100,
and determine its orientation.
8.25 The equation of a particular conic section is
Q≡ 8 x^21 +8x^22 − 6 x 1 x 2 = 110.
Determine the type of conic section this represents, the orientation of its principal
axes, and relevant lengths in the directions of these axes.