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168 ALGEBRA

TABLE 5.2
Types of square matrices (¯a ij is the complex conjugate of a number a ij)

Type of square matrix [a ij] Entries
Unit (identity) a ij = δ ij = 1, i = j, (δ ij is the Kronecker delta)
I =[δ ij] 0, i ≠ j,
any, i = j,

Diagonal a ij =
0, i ≠ j
Upper triangular a ij = any, i ≤ j,
(superdiagonal) 0, i > j
Strictly any, i < j,
upper triangular a ij = 0, i ≥ j
Lower triangular a ij = any, i ≥ j,
(subdiagonal) 0, i < j
Strictly a ij = any, i > j,
lower triangular 0, i ≤ j
Symmetric a ij = a ji (see also Paragraph 5.2.1-3)
Skew-symmetric a ij =–a ji (see also Paragraph 5.2.1-3)
(antisymmetric)
Hermitian a ij = ¯a ji (see also Paragraph 5.2.1-3)
(self-adjoint)
Skew-Hermitian a ij =–¯a ji (see also Paragraph 5.2.1-3)
(antihermitian)
Monomial Each column and each row contain exactly one nonzero entry
(generalized permutation)

A square matrix is a matrix of size n × n,and n is called the dimension of this square
matrix. The main diagonal of a square matrix is its diagonal from the top left corner to
the bottom right corner with the entries a 11 a 22 ... a nn .The secondary diagonal of a
square matrix is the diagonal from the bottom left corner to the top right corner with the
entries a n1 a (n–1)2 ... a 1n . Table 5.2 lists the main types of square matrices (see also
Paragraph 5.2.1-3).

5.2.1-2. Basic operations with matrices.
Two matrices are equal if they are of the same size and their respective entries are equal.
The sum of two matrices A ≡ [a ij ]and B ≡ [b ij ]ofthe same size m × n is the matrix
C ≡ [c ij ]ofsize m × n with the entries

c ij = a ij + b ij (i = 1, 2, ... , m; j = 1, 2, ... , n).
The sum of two matrices is denoted by C = A + B, and the operation is called addition of
matrices.
Properties of addition of matrices:

A + O = A (property of zero),
A + B = B + A (commutativity),
(A + B)+ C = A +(B + C) (associativity),

where matrices A, B, C, and zero matrix O have the same size.