alt="upper A"/> is an
‐by‐
matrix and
is an
‐by‐
matrix. Then
is defined to be the
‐by‐
matrix whose entry in row
, column
, is given by the following equation:
In other words, the entry in row
, column
, of
is computed by taking row
of
and column
of
, multiplying together corresponding entries, and then summing. The number of columns of
must be equal to the number of rows of
.
Example 2.3
then
Definition 2.9 (Square matrix) A matrix
is said to be a square matrix if the number of rows is the same as the number of columns.
Definition 2.10 (Symmetric matrix) A square matrix
is said to be symmetric if
or in matrix notation
all
and
.
Example 2.4
The matrix
is symmetric; the matrix
is not symmetric.
Definition 2.11 (Trace) For any square matrix
, the trace of
denoted by
is defined as the sum of the diagonal elements, i.e.
Example 2.5
Let
be a matrix with
Then
We remark that trace are only defined for square matrices.
Definition 2.12 (Determinant of a matrix) Suppose
is an
‐by‐
matrix,
The determinant of
