Mathematical Methods for Physics and Engineering : A Comprehensive Guide

MATRICES AND VECTOR SPACES

Clearly result (8.63) for diagonal matrices is a special case of this result. Moreover,

it may be shown that the inverse of a non-singular lower (upper) triangular matrix

is also lower (upper) triangular.

8.12.3 Symmetric and antisymmetric matrices

A square matrixAof orderNwith the propertyA=ATis said to besymmetric.

Similarly a matrix for whichA=−ATis said to beanti-orskew-symmetric

and its diagonal elementsa 11 ,a 22 ,...,aNNare necessarily zero. Moreover, ifAis

(anti-)symmetric then so too is its inverseA−^1. This is easily proved by noting

that ifA=±ATthen

(A−^1 )T=(AT)−^1 =±A−^1.

AnyN×NmatrixAcan be written as the sum of a symmetric and an

antisymmetric matrix, since we may write

A=^12 (A+AT)+^12 (A−AT)=B+C,

where clearlyB=BTandC=−CT. The matrixBis therefore called the

symmetric part ofA,andCis the antisymmetric part.

IfAis anN×Nantisymmetric matrix, show that|A|=0ifNis odd.

IfAis antisymmetric thenAT=−A. Using the properties of determinants (8.49) and
(8.51), we have

|A|=|AT|=|−A|=(−1)N|A|.

Thus, ifNis odd then|A|=−|A|, which implies that|A|=0.

8.12.4 Orthogonal matrices

A non-singular matrix with the property that its transpose is also its inverse,

AT=A−^1 , (8.65)

is called anorthogonal matrix. It follows immediately that the inverse of an

orthogonal matrix is also orthogonal, since

(A−^1 )T=(AT)−^1 =(A−^1 )−^1.

Moreover, since for an orthogonal matrixATA=I, we have

|ATA|=|AT||A|=|A|^2 =|I|=1.

Thus the determinant of an orthogonal matrix must be|A|=±1.

An orthogonal matrix represents, in a particular basis, a linear operator that

leaves the norms (lengths) of real vectors unchanged, as we will now show.