502 Chapter 18Matrices and linear transformations
18.2 Some special matrices
Square matrices
If the number of rows of a matrix is equal to the number of columns,m 1 = 1 n, the matrix
is called a square matrixof ordern. For example, a square matrix of order 3 is
(18.11)
In such a matrix, the diagonal containing the elementsa
11, a
22, =,a
nnis called the
principal diagonal(the word ‘principal’ is often omitted), and the elementsa
iion this
diagonal are called the diagonal elementsof the matrix. Those elements,a
ijfori 1 ≠ 1 j,
not on the diagonal are called the off-diagonal elements; the off-diagonal elements of
the matrix (18.11) area
12,a
13,a
23,a
21,a
31,anda
32.
A matrix whose off-diagonal elements are all zero is called a diagonal matrix; for
example,
(18.12)
The diagonal matrix of order nwhose diagonal elements are all unity is called the unit
matrix I(orI
n) of order n. For order 3,
(18.13)
Two important scalar properties of a square matrix Aare the determinant of the
matrix, denoted by|A|or det 1 A,
(18.14)
||AA==detaaa a
aaa a
aaa
nn11
12 13121 22 23231 32 33331 23
a
aaa a
nn nn nnI=
100
010
001
(unit mattrix)
a
a
a
11223300
00
00
(diiagonal matrix)
aaa
aaa
aaa
11
12 1321 22 2331 32 33()square matrix