Matrix Concepts and Basic Operations | 矩阵的概念与基本运算

📚 Matrix Concepts and Basic Operations | 矩阵的概念与基本运算

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is one of the most powerful tools in mathematics, widely used in physics, computer science, economics, and engineering to represent data, solve systems of equations, and describe linear transformations.

矩阵是按行和列排列的矩形数组，其中的元素可以是数字、符号或表达式。矩阵是数学中最强大的工具之一，广泛用于物理、计算机科学、经济学和工程学等领域，用来表示数据、求解方程组以及描述线性变换。

1. Definition and Notation | 矩阵的定义与记号

A matrix with m rows and n columns is called an m × n matrix. It is usually written inside large parentheses or square brackets, with entries denoted by a lowercase letter with double subscripts: aᵢⱼ represents the entry in the i-th row and j-th column.

一个有 m 行 n 列的矩阵称为 m × n 矩阵。矩阵通常写在大括号或方括号内，元素用小写字母加双下标表示：aᵢⱼ 表示位于第 i 行第 j 列的元素。

The general form of an m × n matrix A is written as:

m × n 矩阵 A 的一般形式为：

A = [aᵢⱼ]ₘₓₙ =
[ a₁₁ a₁₂ … a₁ₙ
a₂₁ a₂₂ … a₂ₙ
⋮ ⋮ ⋱ ⋮
aₘ₁ aₘ₂ … aₘₙ ]

The number of rows and columns together is called the order (or dimension) of the matrix. For example, a 2 × 3 matrix has 2 rows and 3 columns.

矩阵的行数和列数合称为矩阵的阶（或维数）。例如，一个 2 × 3 矩阵有 2 行 3 列。

2. Types of Matrices | 矩阵的类型

Several special matrices appear frequently in algebra. Understanding their definitions helps to simplify operations and recognise patterns in problems.

有几类特殊矩阵在代数中经常出现。理解它们的定义有助于简化运算，并在解题中识别模式。

Row matrix | 行矩阵: A 1 × n matrix such as [2 −1 5]. / 一个 1 × n 矩阵，例如 [2 −1 5]。
Column matrix | 列矩阵: An m × 1 matrix such as [3; 0; −2]. / 一个 m × 1 矩阵，例如 [3; 0; −2]。
Square matrix | 方阵: A matrix with equal numbers of rows and columns, e.g. 2 × 2 or 3 × 3. / 行数和列数相等的矩阵，如 2 × 2 或 3 × 3 矩阵。
Zero matrix | 零矩阵: All entries are zero, denoted O. / 所有元素均为零的矩阵，记作 O。
Identity matrix | 单位矩阵: A square matrix with 1s on the main diagonal and 0s elsewhere, denoted Iₙ. / 主对角线元素为 1、其余元素为 0 的方阵，记作 Iₙ。
Diagonal matrix | 对角矩阵: A square matrix where all off-diagonal entries are zero. / 非对角线元素全为零的方阵。

The main diagonal of a square matrix consists of the entries a₁₁, a₂₂, a₃₃, and so on. The identity matrix is the multiplicative identity in matrix algebra.

方阵的主对角线由元素 a₁₁、a₂₂、a₃₃ 等组成。单位矩阵是矩阵代数中的乘法单位元。

3. Matrix Equality | 矩阵相等

Two matrices A and B are equal if and only if they have the same order and every corresponding entry is equal. This condition must hold for all i and j.

两个矩阵 A 和 B 相等，当且仅当它们具有相同的阶，并且每个对应元素都相等。该条件必须对所有 i 和 j 成立。

A = B ⇔ aᵢⱼ = bᵢⱼ for all i, j, and the orders of A and B are identical.

For example, if [x 2; 3 y] = [1 2; 3 4], then x = 1 and y = 4. Matrix equality is often used to set up equations from a single matrix equation.

例如，若 [x 2; 3 y] = [1 2; 3 4]，则 x = 1，y = 4。矩阵相等常用于从单个矩阵方程中建立方程组。

4. Addition and Subtraction of Matrices | 矩阵的加法与减法

Two matrices can be added or subtracted only if they have the same order. The result is obtained by adding or subtracting corresponding entries.

只有同阶矩阵才能进行加法或减法运算。结果矩阵由对应元素相加或相减得到。

If A = [aᵢⱼ] and B = [bᵢⱼ] are both m × n matrices, then:

若 A = [aᵢⱼ] 和 B = [bᵢⱼ] 均为 m × n 矩阵，则：

A + B = [aᵢⱼ + bᵢⱼ], A − B = [aᵢⱼ − bᵢⱼ]

Example: If A = [1 3; 2 0] and B = [4 −1; 3 5], then A + B = [5 2; 5 5] and A − B = [−3 4; −1 −5].

示例：若 A = [1 3; 2 0]，B = [4 −1; 3 5]，则 A + B = [5 2; 5 5]，A − B = [−3 4; −1 −5]。

Addition of matrices is commutative and associative:

矩阵加法满足交换律和结合律：

A + B = B + A, (A + B) + C = A + (B + C)

5. Scalar Multiplication | 数与矩阵的乘法（标量乘法）

When a matrix is multiplied by a real number (scalar) k, every entry of the matrix is multiplied by k.

当一个矩阵乘以一个实数（标量）k 时，矩阵中的每一个元素都要乘以 k。

kA = k[aᵢⱼ] = [k·aᵢⱼ]

For example, if A = [2 −1; 0 3], then 4A = [8 −4; 0 12] and −2A = [−4 2; 0 −6]. Scalar multiplication is distributive:

例如，若 A = [2 −1; 0 3]，则 4A = [8 −4; 0 12]，−2A = [−4 2; 0 −6]。标量乘法满足分配律：

(k + l)A = kA + lA, k(A + B) = kA + kB

Combining matrix addition and scalar multiplication gives the operation of linear combination: c₁A + c₂B. This forms the foundation of the vector space of matrices.

将矩阵加法与标量乘法结合，便得到线性组合运算 c₁A + c₂B。这构成了矩阵向量空间的基础。

6. Matrix Multiplication: Definition | 矩阵乘法：定义

Matrix multiplication is more complex than addition. If A is an m × n matrix and B is an n × p matrix, then the product AB is defined and has order m × p.

矩阵乘法比加法复杂。若 A 是 m × n 矩阵，B 是 n × p 矩阵，则乘积 AB 有定义，且阶为 m × p。

The entry in the i-th row and j-th column of AB is obtained by multiplying each entry of the i-th row of A by the corresponding entry of the j-th column of B and summing the products:

AB 的第 i 行第 j 列元素，由 A 的第 i 行各元素与 B 的第 j 列对应元素相乘后求和得到：

(AB)ᵢⱼ = Σₖ aᵢₖ·bₖⱼ = aᵢ₁b₁ⱼ + aᵢ₂b₂ⱼ + … + aᵢₙbₙⱼ

For the product to exist, the number of columns of A must equal the number of rows of B. This is often called the inner-dimension rule.

乘积要有定义，A 的列数必须等于 B 的行数。这常称为内维数规则。

7. Matrix Multiplication: Worked Example | 矩阵乘法：示例

Let us compute the product C = AB where A is 2 × 2 and B is 2 × 3.

我们来计算乘积 C = AB，其中 A 为 2 × 2，B 为 2 × 3。

A = [1 2; 3 4], B = [2 −1 0; 1 3 5]

The product C has order 2 × 3. We compute each column one by one:

乘积 C 的阶为 2 × 3。我们逐列计算：

C₁₁ = 1×2 + 2×1 = 4; C₁₂ = 1×(−1) + 2×3 = 5; C₁₃ = 1×0 + 2×5 = 10; C₂₁ = 3×2 + 4×1 = 10; C₂₂ = 3×(−1) + 4×3 = 9; C₂₃ = 3×0 + 4×5 = 20.

C₁₁ = 1×2 + 2×1 = 4；C₁₂ = 1×(−1) + 2×3 = 5；C₁₃ = 1×0 + 2×5 = 10；C₂₁ = 3×2 + 4×1 = 10；C₂₂ = 3×(−1) + 4×3 = 9；C₂₃ = 3×0 + 4×5 = 20。

C = AB = [4 5 10; 10 9 20]

Notice that even when both products are defined, AB ≠ BA in general. Matrix multiplication is not commutative.

注意，即使两个乘积都有定义，一般来说 AB ≠ BA。矩阵乘法不满足交换律。

8. Properties of Matrix Multiplication | 矩阵乘法的性质

Matrix multiplication satisfies several important algebraic laws, provided the relevant products and sums are defined.

在有关乘积与和都有定义的条件下，矩阵乘法满足以下重要运算法则。

Associative law | 结合律: (AB)C = A(BC). / (AB)C = A(BC)。
Distributive laws | 分配律: A(B + C) = AB + AC and (A + B)C = AC + BC. / A(B + C) = AB + AC 以及 (A + B)C = AC + BC。
Identity | 单位元: AIₙ = A and IₘA = A for suitable identity matrices. / 对适当的单位矩阵有 AIₙ = A，IₘA = A。
Scalar association | 标量结合: k(AB) = (kA)B = A(kB). / k(AB) = (kA)B = A(kB)。

However, the commutative law does not hold in general: AB ≠ BA. Also, AB = 0 does not imply that A = 0 or B = 0.

但一般而言，交换律不成立：AB ≠ BA。此外，AB = 0 并不能推出 A = 0 或 B = 0。

9. Transpose of a Matrix | 矩阵的转置

The transpose of a matrix A, written Aᵀ or A′, is formed by interchanging rows and columns. If A is m × n, then Aᵀ is n × m, and its entries satisfy (Aᵀ)ᵢⱼ = aⱼᵢ.

矩阵 A 的转置记作 Aᵀ 或 A′，是通过互换行与列得到的。若 A 为 m × n，则 Aᵀ 为 n × m，并且其元素满足 (Aᵀ)ᵢⱼ = aⱼᵢ。

For example, if A = [1 2 3; 4 5 6], then:

例如，若 A = [1 2 3; 4 5 6]，则：

Aᵀ = [1 4; 2 5; 3 6]

Key properties of the transpose include:

转置的重要性质包括：

(Aᵀ)ᵀ = A. / (Aᵀ)ᵀ = A。
(A + B)ᵀ = Aᵀ + Bᵀ. / (A + B)ᵀ = Aᵀ + Bᵀ。
(kA)ᵀ = kAᵀ. / (kA)ᵀ = kAᵀ。
(AB)ᵀ = BᵀAᵀ. The order reverses! / (AB)ᵀ = BᵀAᵀ。乘法顺序要反转！

10. Determinant of a 2 × 2 Matrix | 二阶行列式

For a 2 × 2 matrix A = [a b; c d], the determinant is defined as det(A) = ad − bc. The determinant is a scalar associated with a square matrix and is central to finding inverses.

对于二阶矩阵 A = [a b; c d]，行列式定义为 det(A) = ad − bc。行列式是与方阵相关的标量，在求逆矩阵中至关重要。

det(A) = |A| = ad − bc

If det(A) = 0, the matrix is called singular and has no inverse. If det(A) ≠ 0, the matrix is non-singular and invertible.

若 det(A) = 0，则称矩阵为奇异矩阵，没有逆矩阵。若 det(A) ≠ 0，则矩阵非奇异且可逆。

11. Inverse of a 2 × 2 Matrix | 二阶逆矩阵

For a non-singular 2 × 2 matrix A = [a b; c d], the inverse matrix A⁻¹ is given by:

对于非奇异的二阶矩阵 A = [a b; c d]，其逆矩阵 A⁻¹ 由下式给出：

A⁻¹ = (1/(ad − bc)) × [d −b; −c a]

The inverse satisfies A A⁻¹ = A⁻¹ A = I₂, where I₂ is the 2 × 2 identity matrix.

逆矩阵满足 A A⁻¹ = A⁻¹ A = I₂，其中 I₂ 是二阶单位矩阵。

Example: For A = [2 1; 5 3], det(A) = 2×3 − 1×5 = 1, so A⁻¹ = [3 −1; −5 2].

示例：对于 A = [2 1; 5 3]，det(A) = 2×3 − 1×5 = 1，故 A⁻¹ = [3 −1; −5 2]。

Inverse matrices are used to solve matrix equations of the form AX = B, yielding X = A⁻¹B, provided A is invertible.

逆矩阵用于求解形如 AX = B 的矩阵方程，当 A 可逆时可得 X = A⁻¹B。

12. Using Matrices to Solve Linear Systems | 用矩阵求解线性方程组

A system of linear equations can be expressed compactly in matrix form AX = B. The matrix approach unifies the solving process and is especially useful in higher-dimensional problems.

线性方程组可以简洁地写成矩阵形式 AX = B。矩阵方法统一了求解过程，在高维问题中尤其有用。

Consider the system:

考虑方程组：

2x + 3y = 7
x − y = 1

This system can be written as A X = B with A = [2 3; 1 −1], X = [x; y], B = [7; 1].

该方程组可写成 A X = B，其中 A = [2 3; 1 −1]，X = [x; y]，B = [7; 1]。

Since det(A) = 2×(−1) − 3×1 = −5 ≠ 0, the inverse exists:

因为 det(A) = 2×(−1) − 3×1 = −5 ≠ 0，所以逆矩阵存在：

A⁻¹ = (1/−5) × [−1 −3; −1 2] = [0.2 0.6; 0.2 −0.4]. Then X = A⁻¹B = [2; 1]. Hence x = 2 and y = 1.

A⁻¹ = (1/−5) × [−1 −3; −1 2] = [0.2 0.6; 0.2 −0.4]。因此 X = A⁻¹B = [2; 1]，故 x = 2，y = 1。

This method shows the power of matrix algebra: instead of manipulating equations directly, one obtains all solutions simultaneously through one matrix product.

此方法展示了矩阵代数的威力：无需逐个变形象限方程，只需一次矩阵乘法即可同时获得所有解。

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Matrix Concepts and Basic Operations | 矩阵的概念与基本运算