Matrix theory basics 2.1 Basic concepts of a Matrix

When finding linear equations, the coefficients and their respective locations are very important. When changing, the order of the equations must be maintained, therefore, we often arrange the coefficients of the unknown in a number table of a rectangle.

For example, Linear Equations

The coefficients are arranged in a rectangle number table according to the original positions of the equations:

,

We call this rectangle number table as a matrix.

Definition 1M'NNumberAij(I= 1, 2,
×
×
×
,
M;
J= 1, 2,
×
×
×,
N) ArrangedMLineNColumn rectangle number table

CalledM'NLevel matrix. Generally, we use uppercase letters.A,
B,
CAnd mark it

,

In the matrixM'NA number is called a matrix element. All elements and their corresponding positions are integral. Therefore, you must add square brackets to indicate it.

M x n tuples in matrix A [a11, a12 ,..., A1n], [a21, a22 ,..., A2n],..., [Am1, am2 ,..., Amn] is called a matrix row; n vertical m tuples

A column called a matrix.

AijTheIRowJColumn element. A hasM'NElements, soM'NMatrixAIt can also be expressedA= (Aij)M'NOr recordedA
M'N. For example

Indicates a matrix with two rows, three columns, and six elements.

 

Indicates a matrix with three rows, four columns, and 12 elements.

Indicates a matrix with two rows, two columns, and four elements.

If MatrixAThe number of rows and columns are equalNAIsNLevel matrix, or calledNSquare matrix.
NLevel MatrixAAlso recordedAn
.

There is only one difference between the rank-n matrix and the rank-n deciding factor in form. The former is to enclose the square number of n in parentheses and is a number table, the latter uses two vertical bars to enclose the square number of n, with n! The complex expansion of an item. It is a definite number after it is substituted into a specific element value.

Only when the matrix of a row is m = 1,A= (A1A2 ×
×
×
AnIs called the n-element row matrix, or an n-dimensional row vector. The row matrix is also recorded

A= (A1,
A2,
×
×
×
,
An).

Matrix with only one column

It is called the m-element column matrix or m-dimensional column vector.

When all elements in a matrix are real numbers, they are called real matrices. A matrix whose elements are 0 is called a zero matrix, which is recordedO.

The two matrices have the same number of rows and number of columns.A= (Aij) AndB= (Bij) Is the same type matrix, and their corresponding elements are equal, that is

Aij=Bij(I= 1, 2,
×
×
×,
M;
J= 1, 2,
×
×
×,
N),

Weighing MatrixAAnd MatrixBEqual, recordedA=B.

For example, if

Then a = 1, B = 3, c = 5, d = 7.

Note: The zero Matrices of different types are not equal.

When studying the linear transformation between variables, linear transformation can correspond to the matrix.

SetNVariable (s)X1,
X2,
×
×
×,
XnAndMVariable (s)Y1,
Y2,
×
×
×,
YmRelationship

Indicates a slave variable.X1,
X2,
×
×
×,
XnTo variableY1,
Y2,
×
×
×,
YmLinear transformation, whereAijConstant. linear transformation coefficientAijMatrixA= (AIJ)M'N, That is

It is called a coefficient matrix.

Given a linear transformation, the matrix composed of its coefficients is also determined. if a matrix is given as a linear transformation coefficient matrix, the linear transformation is also determined. in this sense, there is a one-to-one correspondence between the linear transformation and the matrix. matrix A can reflect the characteristics of linear transformation.

If it corresponds to a constant transformation

It corresponds to a Coefficient MatrixNSquare Matrix

.

This square matrix is characterized by the fact that all elements on the main diagonal line are 1 and all other elements are 0. We call this matrixNLevel unit matrix (unit matrix) is usually recorded as En. On the contrary, the linear transformation corresponding to a unit matrix must be a constant transformation.

Linear transformation

CorrespondingNSquare Matrix

.

This type of matrix is called a diagonal matrix.

L = diag (L1,
L2,
×
×
×,
Ln).

If the variable group is regarded as an m-column matrix, the variable group is considered as an n-element column matrix. linear transformation can be expressed as Y = AX by multiplication of the matrix.

Similarly, the linear equations can be considered

 

AX = B

,

AIt is called a coefficient matrix.

We will discuss the matrix in the next chapter.AThe relationship between the nature and the Solution of Linear Equations

System.