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Algebra of matrice

Definition: A matrix A over a field K or, simply, a matrix A is a regular array of scalars usually presented in the following form

matrix-1

The rows of such a matrix A are the m horizontal lists of scalars:

matrix-2

and the terms of A are the n verticals lists of scalars:

matrix-3matrix-4

Matrix addition and scalar multiplication:

 
be two matrices with the same size, say m×n matrices. The sum of A and B, written A+B, is the matrix obtained by adding corresponding elements from A and B. That is

matrix-5

 

The product of the matrix A by a scalar k, written k.A or simply kA, is the matrix obtained by multiplying each element of A by k. That is,

matrix-6

Matrix multiplication:
The product of matrix A and B written AB, is some what complicated. For this reason, we first begin with a special case.
The product AB of a row matrix

matrix-7

with the same number of elements is defined to be the scalar obtained by multiplying corresponding entries and adding; that is

matrix-8

matrix-9

Transpose of a matrix: The transpose of a matrix A, written AT, is the matrix obtained by writing the columns A, in order, as rows.
For example,

matrix-10

In other words, if A = [aij] is an m×n matrix, then AT=[bij] is the n×m matrix where bij=aji

Square matrices

A square matrix is a matrix with the same number of rows as columns. An n×n square matrix is said to be of order n and is sometimes called an n – square matrix.

Diagonal and Trace

Let A = [aij] be an square matrix. The diagonal or main diagonal of A consists of the elements with the same subscripts, that is,

matrix-11

The trace of A, written tr(A), is the sum of the diagonal elements. Namely,


matrix-12

Identity matrix, Scalar matrices
The n-square identity or unit matrix, denoted by In, or simply I, is the n- square matrix with I’s on the diagonal and 0’s elsewhere. The identity matrix I is similar to the scalar 1 in that, for any n- square matrix.

matrix-13

 

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