Data Science in Theory and Practice. Maria Cristina Mariani

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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

, denoted det

or

, is defined by

      where

are referred to as the “cofactors” and are computed from

      The term

is known as the “minor matrix” and is the matrix you get if you eliminate row

and column

from matrix

.

Finding the determinant depends on the dimension of the matrix

; determinants only exist for square matrices.

       Example 2.6

      For a 2 by 2 matrix

      we have

       Example 2.7

      For a 3 by 3 matrix

      we have

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