Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(lu) #1

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=


iαie

i.Hereμis a given constant andvis a given column
matrix.

(a) Solve (∗)when

A=




210


120


003



,


μ=2andv=(123)T.
(b) Would (∗) have a solution ifμ=1and(i)v=(123)T, (ii)v=
(223)T?
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