Page 89 - Matrix Analysis & Applied Linear Algebra

3.2 Addition and Transposition 83

Properties of Scalar Multiplication
For m × n matrices A and B and for scalars α and β, the following
properties hold.
Closure property: αA is again an m × n matrix.

Associative property: (αβ)A = α(βA).

Distributive property: α(A + B)= αA + αB. Scalar multiplica-
tion is distributed over matrix addition.
Distributive property: (α + β)A = αA + βA. Scalar multiplica-
tion is distributed over scalar addition.

Identity property: 1A = A. The number 1 is an identity el-
ement under scalar multiplication.

Other properties such as αA = Aα could have been listed, but the prop-
erties singled out pave the way for the definition of a vector space on p. 160.
A matrix operation that’s not derived from scalar arithmetic is transposition
as defined below.

Transpose
The transpose of A m×n is defined to be the n × m matrix A T ob-
tained by interchanging rows and columns in A. More precisely, if
T
A =[a ij ], then [A ] ij = a ji . For example,
  T
12
 34  = 135 .
246
56
T T
It should be evident that for all matrices, A = A.

Whenever a matrix contains complex entries, the operation of complex con-
jugation almost always accompanies the transpose operation. (Recall that the
complex conjugate of z = a +ib is defined to be z = a − ib.)