Matrix matrix
1. Matrix matrix
1.1. Definitions
number table by the number m of m rows × n aij arranged in a matrix of n columns referred to as m rows and n columns, referred to as m × n matrix. Referred to as:
This number m × n matrix A is called the element, referred to as the element, the number of aij i-th row j-th column of the matrix A, the matrix A is called (i, j) element, as the number of aij (i, j ) elements may be referred to as the matrix (of aij) or (aij) m × n, m × n matrix a is also referred to as Amn.
Matrix elements are real numbers referred to as a real matrix, the matrix elements are called complex matrix complex. The number of rows and columns equal to n matrix is called matrix of order n or n-order square matrix [8].
A ∈ Rm × n symbol represents a matrix of m rows and n columns, and the matrix A are all elements are real numbers.
The symbol represents a vector x ∈ Rn containing n elements. Typically, we viewed as a n-dimensional vector a matrix of n rows, i.e., column vector. If we want to represent a vector row (row n-column matrix 1), we usually written xT (xT denotes the transpose of x, which is defined as explained later).
A use or aj:, j represents the j-th column element of the matrix A:
Or Ai ,: with the aT i represents the i-th row matrix elements:
1.2 Basic operation
The basic operation matrix comprises a matrix of addition, subtraction, multiplication, transpose, conjugate and conjugate transpose.
addition
Adder matrix satisfies the following calculation law (A, B, C are the same matrix type):
It should be noted that it can only be an addition between the same type of matrix.
Subtraction
Multiplication
The matrix multiplication operation to meet the following law:
Subtraction and multiplication of a matrix collectively linear operation matrix [8].
Transpose
the transposed matrix the rows and columns of the matrix A generated by exchange matrix known as the A (), a process known as the transposed matrix
Matrix transpose operation satisfies the following law:
Multiplication
Multiplication of two matrices can be defined only if the number of rows equal to the number of columns of a first matrix A and matrix B of another. As A is an m × n matrix and B is an n × p matrix, and their product is a C m × p matrix
, one of its elements:
This product will be written as:
E.g:
Matrix multiplication operations to meet the following law:
Associative law:
Left distributive law:
Right distributive law:
Matrix multiplication is not commutative.