MATRICES AND VECTOR SPACES
(b) find an orthonormal basis, within a four-dimensional Euclidean space, for
thesubspacespannedbythethreevectors(1200)T,(3 −120)T
and(0021)T.8.14 If a unitary matrixUis written asA+iB,whereAandBare Hermitian with
non-degenerate eigenvalues, show the following:
(a) AandBcommute;
(b)A^2 +B^2 =I;
(c) The eigenvectors ofAare also eigenvectors ofB;
(d) The eigenvalues ofUhave unit modulus (as is necessary for any unitary
matrix).8.15 Determine which of the matrices below are mutually commuting, and, for those
that are, demonstrate that they have a complete set of eigenvectors in common:
A=
(
6 − 2
− 29
)
, B=
(
18
8 − 11
)
,
C=
(
− 9 − 10
− 10 5
)
, D=
(
14 2
211
)
.
8.16 Find the eigenvalues and a set of eigenvectors of the matrix
13 − 1
34 − 2
− 1 − 22
.
Verify that its eigenvectors are mutually orthogonal.
8.17 Find three real orthogonal column matrices, each of which is a simultaneous
eigenvector of
A=
001
010
100
and B=
011
101
110
.
8.18 Use the results of the first worked example in section 8.14 to evaluate, without
repeated matrix multiplication, the expressionA^6 x,wherex=(2 4 −1)Tand
Ais the matrix given in the example.
8.19 Given thatAis a real symmetric matrix withnormalised eigenvectorsei,obtain
the coefficientsαiinvolved when column matrixx, which is the solution of
Ax−μx=v, (∗)is expanded asx=∑
iαiei.Hereμis a given constant andvis a given column
matrix.(a) Solve (∗)whenA=
210
120
003
,
μ=2andv=(123)T.
(b) Would (∗) have a solution ifμ=1and(i)v=(123)T, (ii)v=
(223)T?