Fundamentals of Matrix Algebra 387
Note that the first subscript indicates rows while the second subscript
indicates columns. The entries aij—called the elements of the matrix A—are
the numbers at the crossing of the ith row and the jth column. The commas
between the subscripts of the matrix entries are omitted when there is no
risk of confusion: aaij, ≡ ij. A matrix A is often indicated by its generic ele-
ment between brackets:
A={}aij nm or A=aijnmwhere the subscripts nm are the dimensions of the matrix.
There are several types of matrices. First there is a broad classification
of square and rectangular matrices. A rectangular matrix can have differ-
ent numbers of rows and columns; a square matrix is a rectangular matrix
with the same number n of rows as of columns. Because of the important
role that they play in applications, we focus on square matrices in the next
section
Square Matrices
The n × n identity matrix, indicated as the matrix In, is a square matrix in
which diagonal elements (i.e., the entries with the same row and column
suffix) are equal to one while all other entries are zero:
=
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅⋅
⋅ ⋅⋅⋅
⋅⋅ ⋅⋅
⋅⋅⋅
I
10 0
01 0
00 1
nA matrix in which entries are all zero is called a zero matrix.
A diagonal matrix is a square matrix in which elements are all zero
except the ones on the diagonal:
=
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅⋅
⋅⋅ ⋅⋅
⋅⋅ ⋅⋅
⋅⋅⋅
a
aaA
00
00
(^00) nn
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