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Matrix Definition. Main information

Matrix definition

Definition.
Matrix with size n×m is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns, which consisting of n rows and m columns.

The number of rows and columns are defined the matrix size.


Matrix notation

Matrices are commonly written in box brackets or large parentheses:

A = ( 4  1  -7 )
 -1  0  2 

The matrix usually denoted by capital letters of the Latin alphabet. The matrix comprising n rows and m columns, called the matrix of size n×m. Also, the size of the matrix is written following way: An×m.


Matrix entries

Entries of the matrix A are denoted aij, where i is row number, in which is an element, j is the column number.
Example.
Entries of matrix A4×4:
A = ( 4  1  -7  2 )
 -1  0  2  44 
 4  6  7  9 
 11  3  1  5 

a11 = 4

Definition.
Matrix row is zero if all of its elements equal to zero.
Definition.
If at least one of the elements of the matrix row is not zero, the row is called a non-zero.
Example.
Zero and non-zero matrix row:

( 4  1  -7 )
< non-zero matrix row
 0  0  0 
 0  1  0 
Definition.
Matrix column is zero if all of its elements equal to zero.
Definition.
If at least one of the elements of the matrix column is not zero, the column is called a non-zero.
Example.
Zero and non-zero matrix column:

( 0  1  -7 )
 0  0  2 
^
^
^

non-non-zero matrix column


Matrix diagonal

Definition.
Main diagonal of matrix is the collection of entries aij where i = j.
Definition.
Antidiagonal of matrix with size n×m is the collection of entries aij where i + j = n + 1.
Example.
Main diagonal and antidiagonal of matrix:

( 0  1  -7 ) - main diagonalantidiagonal of matrix
 0  0  2 

( 0  1  -7 ) - main diagonalantidiagonal of matrix
 0  0  2 
 8  2  9 
Definition.
The trace of square matrix A is defined to be the sum of the elements on the main diagonal.
Definition.
The trace of matrix is denoted as

tr(A) = a11 + a22 + ... + ann.

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