A
system of linear equations
can easily be written with a matrix. This allows us to make the manipulations which lead to the solution of such a system, without writing down the variables. Matrices are quite simple objects; however, they can represent quite different mathematical objects
(e.g., a family of column vectors, a family of row vectors, a linear mapping, a table of physical interactions, a vector field, etc.),
which one has to keep in mind in order to prevent wrong conclusions.
Let denote a
field,
and let
and
denote index sets. An -matrix is a
mapping
-
If
and
,
then we talk about an -matrix. In this case, the matrix is usually written as
-
We will usually restrict to this situation. For every
,
, ,
is called the -th row of the matrix, which is usually written as a row vector
-
For every
,
, ,
is called the -th column of the matrix, usually written as a column vector
-
The elements are called the entries of the matrix. For , the number is called the row index, and is called the column index of the entry. The position of the entry is where the -th row meets the -th column. A matrix with
is called a square matrix. An -matrix is simply a column tuple
(or column vector)
of length , and an -matrix is simply a row tuple
(or row vector)
of length . The set of all matrices with rows and columns
(and with entries in )
is denoted by , in case
we also write .
Two matrices
are added by adding entries with corresponding entries. The multiplication of a matrix with an element
(a scalar) is also defined entrywise, so
-
and
-
The multiplication of matrices is defined in the following way.
Such a matrix multiplication is only possible when the number of columns of the left-hand matrix equals the number of rows of the right-hand matrix. Just think of the scheme
-
the result is an -Matrix. In particular, one can multiply an -matrix with a column vector of length
(the vector on the right),
and the result is a column vector of length . The two matrices can also be multiplied with roles interchanged,
-
An
-matrix
of the form
-
is called a
diagonal matrix.
The
-matrix
-
is called
identity matrix.
The identity matrix has the property
,
for an arbitrary -matrix .