Anon

Fundamentals of Matrix Algebra 393

Multiplication We define two types of multiplication: (1) multiplication of
a scalar and a vector and (2) scalar multiplication of two vectors (inner
product).^2
The multiplication of a scalar a and a row (or column) vector x, denoted
as ax, is defined as the multiplication of each component of the vector by
the scalar:

aax=[]xa 1 ,..., xn

A similar definition holds for column vectors. It is clear from this definition
that multiplication by a scalar is associative as

aa()xy+=xy+a

The scalar product (also called the inner product) of two vectors x, y,
denoted as xy⋅ , is defined between a row vector and a column vector. The
scalar product between two vectors produces a scalar according to the fol-
lowing rule:

xy⋅ =

=

∑xyii

i

n

1

Two vectors x, y are said to be orthogonal if their scalar product is zero.

Matrix Operations

Let’s now define operations on matrices. The following five operations on
matrices are usually defined: transpose, addition, multiplication, inverse,
and adjoint.

Transpose The definition of the transpose of a matrix is an extension of the
transpose of a vector. The transpose operation consists in exchanging rows
with columns. Consider the n × m matrix

A={}aijnm

(^2) A third type of product between vectors—the vector (or outer) product between
vectors—produces a third vector. We do not define it here as it is not typically used
in economics though widely used in the physical sciences.