Data Science in Theory and Practice. Maria Cristina Mariani

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