8.12 SPECIAL TYPES OF SQUARE MATRIX
leading diagonal, i.e. only elementsAijwithi=jmay be non-zero. For example,
A=10 0
02 0
00 − 3,is a 3×3 diagonal matrix. Such a matrix is often denoted byA= diag (1, 2 ,−3).
By performing a Laplace expansion, it is easily shown that the determinant of an
N×Ndiagonal matrix is equal to the product of the diagonal elements. Thus, if
the matrix has the formA= diag(A 11 ,A 22 ,...,ANN)then
|A|=A 11 A 22 ···ANN. (8.63)Moreover, it is also straightforward to show that the inverse ofAis also a
diagonal matrix given by
A−^1 = diag(
1
A 11,1
A 22,...,1
ANN)
.Finally, we note that, if two matricesAandBarebothdiagonal then they have
the useful property that their product is commutative:
AB=BA.This isnottrue for matrices in general.
8.12.2 Lower and upper triangular matricesA square matrixAis calledlower triangularif all the elementsabovethe principal
diagonal are zero. For example, the general form for a 3×3 lower triangular
matrix is
A=A 11 00
A 21 A 22 0
A 31 A 32 A 33,where the elementsAijmay be zero or non-zero. Similarly anupper triangular
square matrix is one for which all the elementsbelowthe principal diagonal are
zero. The general 3×3 form is thus
A=A 11 A 12 A 13
0 A 22 A 23
00 A 33.By performing a Laplace expansion, it is straightforward to show that, in the
generalN×Ncase, the determinant of an upper or lower triangular matrix is
equal to the product of its diagonal elements,
|A|=A 11 A 22 ···ANN. (8.64)