Types of Matrices

Quadratic Matrix is ​​a matrix whose number of rows is equal to the number of columns.


The zero matrix is ​​a matrix with all entries equal to zero

Matrix properties zero:

A + 0 = A (if the size of the matrix A = matrix size 0)
A0 = 0; 0A = 0 (if multiplication requirements are met)

The Diagonal Matrix is ​​a quadratic matrix where all of the entries outside the main diagonal are zero.


Identity Matrix or Unit Matrix is a diagonal matrix whose main diagonal entries are all numbers 1. The ordinary identity matrix is ​​written I or In where n shows the size (order) of the quadratic matrix.

The nature of the identity matrix is ​​like the number 1 (one) in operations with ordinary numbers, namely:

AI = A; IA = A (if conditions are met)

Example 1
Then, 

The lower triangular matrix is ​​a quadratic matrix with all entries above diagonal = 0.


An upper triangular matrix is ​​a quadratic matrix in which all the entries below the main diagonal = 0

Symmetrical Matrix is ​​a matrix whose transposes are the same as themselves, in other words if A = At.

Because A = At., Then A is a symmetrical matrix.

Antisymmetric Matrix is ​​a matrix whose transposes are negative or At = - A. It is easy to understand that all the main diagonal entries of the antisymmetric matrix are 0.


Because At = - A, the matrix A is an antisymmetric matrix.

Commutative Matrix
If A and B are squared metrics and apply AB = BA, then A and B are said to be communicative with each other. It is clear that each commutative square I (which is the same size) and with its inverse (if any). If AB = -BA, it is said to be anticomutative.alau AB = -BA, dikatakan antikomutatif.


Idempotent Matrix, Periodic, Nilpoten
The quadratic matrix A is said to apply the bile Idempotent Matrix AA = A2 = A. Generally if p is the smallest (positive round) number, so AA applies ... A = Ap = A, then it is said A Periodic Matrix with period (p - 1) . If Nilpoten A is At = 0 for an natural number r it is said A is nilpotent with index r. The Matrix Index is the smallest positive integer r that satisfies the Ar = 0 relationship.


Because, 

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