[Serial] Notes on Linear Algebra-Chapter 2 Matrix

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Textbook: Linear Algebra (Third Edition)-Shanghai Jiaotong University Press

Chapter 2 | Matrix

The matrix is ​​not a determinant, it does not represent a certain value, but the real one is a table of numbers

The system of linear equations can be viewed as:

• Coefficient matrix
• Augmented matrix-the sum + coefficient matrix of each equation in the linear equation system

Use the method of addition, subtraction and elimination to solve the equations:

• The concept of = does not exist in the matrix. Because the matrix changes during the transformation process, the corresponding number table changes, so arrows can only be used.
• Although the corresponding equations are the same solution, the matrix is ​​changed (enlarged matrix)

2.1 Matrix concept

A rectangular number table with m×n numbers, m rows and n columns, enclosed in parentheses (long brackets) on both sides is called a matrix.

Writing aij, i, j=1, 2, 3...m;

Marked as A, B, C;

There are real R matrices and complex i matrices (both real and imaginary numbers, both real and imaginary parts), and even polynomial matrices

1. Linear operation

Homogeneous matrix

A = （aij） m × n ， B = （at） s × t

When m=s, n=t, A and B are the same type matrix

The equality of two matrices

When aij=bij, and A and B are the same type matrix, A=B, the two matrices are equal.

Special type

• Row (column) matrix

  • Only one row (column);

  • Means:

    • Figure!
    • • Row vector, column vector
      • Line: Do not add a comma between each element, just open it directly

• Zero matrix

  • Elements are all zero
  • Any two zero matrices are not necessarily equal-they must be of the same type.
  • Means:
    • （）m×n

• phalanx

  • Means:

    • Figure!

  • The main diagonal of the square matrix

  • Main diagonal element

  • Determinant of a square matrix

    • $$|A|=|aij|n$$

    • |aij|n represents a value

• diagonal matrix

  • Only the main diagonal has elements.

  • Means:

    • 【Aij = 0, i! = J, i, j = 1,2 …… n】

    • Since there are too many zeros, it is abbreviated as

      • diag (a11, a22,...ann (diagonal elements) (plus comma))

• Quantity matrix

  • It is a diagonal matrix.

  • Means:

    -diag（a,a,a…a）

• Identity matrix

  • diag(1,1,1…1)
  • The diagonal elements are all 1, => is marked as E or I.
  • En=》n-order unit matrix

• Triangular matrix

  • Upper and lower triangle

• Symmetric and Antisymmetric Matrices

  • Symmetric matrix—take the main diagonal as the axis, aij=aji, i, j=1, 2, 3...n;

    • A=AT；

  • Anti-symmetric matrix-the main diagonal elements are all zero, aij=-aji, i, j=1, 2...n;

2.2 Matrix operations

The calculation is divided into:

Calculation
Linear	Non-linear
Addition, multiplication (a total of 8 properties)	AB=C, matrix multiplication

Matrix multiplication
Not satisfied	Satisfy
AB! = BA, commutative
1，2，3…

mind Mapping. . .

1. Addition

​ Negative matrix, it must be clear who is relative to;

-A=（-aij）m×n
-A是数表-，-aij是每个元素-。

A=（aij）m×n，B=（bij）m×n；（两个同型矩阵才可以相加噢~）

• （A + B） = （aij + at） m × n

2. Subtraction

AB = A + (- B) = （aij-at） m × n ；

The nature of the operation is as follows:

A+B=B+A;[加法交换律]
(A+B)+C=A+(B+C);[加法结合律]
A+0=0+A=A;[有零元；0是零矩阵；推论：若某数表+x=x+某数表=某数表，则x为零元]
A+(-A)=0；[有负元]

3. Multiplying the Matrix

​A=(aij)m×n;
cA=(caij)m×n; [c is a number]
-A=(-1)A;
cE=diagonal 1*c

Operational nature:

1A=A;[数乘有单位]
k（lA）=（kl）A;矩阵结合律
数字加法：（k+l）A=kA+lA；[矩阵关于数乘的分配律]
矩阵加法:k(A+B)=kA+kB;[数乘关于矩阵的分配律]
0A=O;[零元]
（-1）A=-A；
k0=0；
若kA=0，则k=0或A=0；

Matrix multiplication

There is no exchange law, no exchange!

Multiplying A and B = C;

• Am × s × Bs × n = Cm × n;
  Immediate 1, column 2 = C12.

A的列数=B的行数
C的行数=A的行数
C的列数=B的列数

Definition :

A·B=(cij)m×n=C——A=（aij）m×p，B=（bij）p×n；
cij=ai1b1j+ai2b2j+......+aipbpj=求和公式（P，k=1）aikbkj；i=1，2....m;j=1,2.....n。

AB!=BA, A·B=C, call C the product of A left multiplied by B (B right multiplied by A).

• Am×nEn=A, Emam×n=A (matrix A multiplied by n identity matrix is ​​still itself).

  • 3×2 column matrix×E2 second-order unit matrix;
  • E3 third order unit matrix×3×2 order matrix;

• Am×nOn×p=Om×p；

• Os×mAm×n=Os×n。

• Non-zero matrix×a matrix=0 matrix, AB!=BA.

  • Can't push each other~

If the two matrices AB satisfy AB=BA, then the matrices A and B can be exchanged.

若两矩阵可交换，则称两矩阵为同阶的方阵。 

As×mBm×s=（）s；
Bm×sAs×m=（）m；
所以m=s，所以A、B为方阵。

6. Does not meet the * commutative law

Unit matrix

The identity matrix can be exchanged with any square matrix of the same order.
EnAnn=An×nEn=An×n.

Quantity matrix

The quantity matrix is ​​interchangeable with any square matrix of the same order.

B为n阶数量矩阵，A为n阶矩阵。
所以B=cEn，（cEn）An×n=An×n（cEn）【Ann】=cA。

------------------------------------The identity matrix can be omitted~-------- ------------------------

Am×n[square matrix]On×p=Om×p[zero matrix], Os×mAm×n=Os×n[zero matrix]

[If AB=O, A=O/B=O cannot be inferred];
[If AB=CB or BA=BC, then A=C is not necessarily true]; the elimination law is not satisfied.
Certificate=》(AC)B=0—!>AC=0-----!>A=C

Satisfy matrix multiplication

- 结合律：（AB）C=A(BC)[顺序不能变]
- 分配律：
		- （A+B）C=AC+BC
		- A(C+D)=AC+AD
- kAB=(kA)B=AkB；

Use matrix operations to express linear equations

[Non-square, equation n [number of rows]! = Unknown number n [number of columns]]
vector representation: graph!

A coefficient m×n matrix, X=unknown quantity column matrix, B=constant column matrix.

A (row) X (column) = B [row] (Ax=b);

A=（aij）3×3，对任意3阶矩阵B都可交换，AB=BA,则可证明A是数量矩阵！【结论牢记~】

Examples of trace tr proof questions:
Picture!

11. display method*

Square power

Suppose A is a square matrix of order n, you must:

A^2=A·A
A^3=A^2*A=A*A*A
A^n=A^(n-1)*A
规定A^0=En单位矩阵；

Polynomial of square matrix A

f（A）=a0A^m +a1A^(m-1）+.....+amE（a1、a2....am为已知常数）
An阶方阵，Enn阶单位阵
方阵f（A）是由f（x）生成的矩阵多项式。
f（x）=a0X^m +a1X^(m-1）+.....+am；

Exponentiation properties

A^m*A^n=A^(m+n);m,n为非负整数
（A^m）^n=A^(mn),A为方阵

Generally, let A and B be square matrices of the same order,
(AB)^m不一定=A^mB^n
because (AB)^2=ABAB！=A^2B^2[not interchangeable]
, the inference:
Picture!

If
it can be exchanged , both are true.
Figure!

Mathematical induction

B^n=Picture!

Matrix transpose

Nature:
Figure!

Symmetric and asymmetric matrices

The properties of determinant operations of square matrices

n-order matrix.

1. 性质：
	2. 
3. 

Conjugate operation

1. 实矩阵与复矩阵
2.设A为复矩阵【实部加虚部】，则A=（aij）m×n
称矩阵（aij的共轭）m×n为A的共轭矩阵。

The nature of the operation of complex numbers
Figure!

pending upgrade…

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It was first written on January 21st, 2021/3.
Update on 2021/3/28.