Matrices · 矩阵
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| matrix/ˈmeɪtrɪks/ | 矩阵 | jǔ zhèn |
| transformation/trænsfɔːˈmeɪʃn/ | 变换 | biàn huàn |
| identity matrix/aɪˈdentɪti ˈmeɪtrɪks/ | 单位矩阵 | dān wèi jǔ zhèn |
| determinant/dɪˈtɜːmɪnənt/ | 行列式 | háng liè shì |
| inverse/ɪnˈvɜːs/ | 逆 | nì |
| singular/ˈsɪŋɡjʊlə/ | 奇异 | qí yì |
| invariant/ɪnˈveərɪənt/ | 不变 | bù biàn |
| enlargement/enˈlɑːdʒmənt/ | 放大 | fàng dà |
The mathematics that powers computer graphics
- Every time you rotate an image on your phone, a matrix 矩阵 multiplication is doing the work behind the scenes.
- Matrices are the language of transformations 变换 — they describe rotations, reflections, stretches, and projections in a single compact object.
驱动计算机图形的数学
- 每当你在手机上旋转一张图像,一个矩阵乘法就在幕后做这件事。
- 矩阵(matrices)是变换的语言——它们用一个紧凑的对象描述旋转、反射、拉伸和投影。
Matrix basics
- A matrix is a rectangular block of numbers. Multiply row × column.
- The identity matrix 单位矩阵 $I$ leaves a matrix unchanged: $AI = IA = A$.
Worked example. $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \\ 6 \end{pmatrix} = \begin{pmatrix} 1(5) + 2(6) \\ 3(5) + 4(6) \end{pmatrix} = \begin{pmatrix} 17 \\ 39 \end{pmatrix}$.
矩阵基础
- 一个矩阵是一个矩形的数块。乘法是 行 × 列。
- 单位(identity)矩阵 $I$ 让一个矩阵不变:$AI = IA = A$。
算例。 $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 \\ 6 \end{pmatrix} = \begin{pmatrix} 1(5) + 2(6) \\ 3(5) + 4(6) \end{pmatrix} = \begin{pmatrix} 17 \\ 39 \end{pmatrix}$。
Matrices · 矩阵
(x', y') = M(x, y)
A matrix transforms the plane; its determinant is the area scale factor (negative = a flip). · 一个矩阵变换平面;它的行列式是面积比例因子(负的 = 一次翻转)。
Multiply [[1, 2], [3, 4]] by [[5], [6]]. The top entry of the result is? · 把 [[1, 2], [3, 4]] 乘以 [[5], [6]]。结果的顶部元素是多少?
1(5) + 2(6) = 5 + 12 = 17. · 1(5) + 2(6) = 5 + 12 = 17。
Put the steps for finding the inverse of a 2x2 matrix [[a,b],[c,d]] in order. · 把求一个 2x2 矩阵 [[a,b],[c,d]] 的逆的步骤按顺序排列。
The inverse is (1/det)[[d,−b],[−c,a]] — swap the diagonal, negate the off-diagonal, divide by det. · 逆是 (1/det)[[d,−b],[−c,a]]——交换对角线,把非对角线取负,除以 det。
Determinants 行列式 and inverses 逆
- For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$: the determinant is $\det A = ad - bc$.
- If $\det A \neq 0$ (non-singular 奇异), the inverse is $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
Singular matrices have no inverse. If $\det A = 0$, the matrix is singular — it squashes the plane into a line, losing information that can't be recovered.
行列式与逆矩阵
- 对 $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$:行列式(determinant)是 $\det A = ad - bc$。
- 如果 $\det A \neq 0$(非奇异),逆矩阵(inverse)是 $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$。
奇异矩阵没有逆。 如果 $\det A = 0$,矩阵是奇异的(singular)——它把平面压成一条线,丢失了无法恢复的信息。
What is the determinant of the matrix [[3, 1], [2, 4]] (ad − bc)? · 矩阵 [[3, 1], [2, 4]] 的行列式(ad − bc)是多少?
det = 3×4 − 1×2 = 12 − 2 = 10. · det = 3×4 − 1×2 = 12 − 2 = 10。
A 2×2 matrix has an inverse exactly when its determinant is: · 一个 2×2 矩阵恰好在它的行列式是以下时有逆:
A non-singular matrix (det ≠ 0) is invertible; det = 0 means singular (no inverse). · 一个非奇异矩阵(det ≠ 0)可逆;det = 0 意味着奇异(没有逆)。
For A = [[3, 1], [2, 4]] with det = 10, the top-left entry of A⁻¹ is d/det. Find it. · 对 A = [[3, 1], [2, 4]],det = 10,A⁻¹ 的左上角元素是 d/det。求它。
A⁻¹ = (1/10)[[4, -1], [-2, 3]]. Top-left = 4/10 = 0.4. · A⁻¹ = (1/10)[[4, -1], [-2, 3]]。左上角 = 4/10 = 0.4。
Matrices as transformations
- A $2\times2$ matrix represents a transformation of the plane; $\det A$ = the area scale factor.
A 2x2 matrix maps the unit square to a parallelogram; its determinant is the area scale factor.
- A product $AB$ means "do $B$, then $A$". Points/lines that don't move are invariant 不变.
矩阵作为变换
- 一个 $2\times2$ 矩阵代表平面的一个变换(transformation);$\det A$ = 面积比例因子。
一个 2x2 矩阵把单位正方形映射到一个平行四边形;它的行列式是面积比例因子。
- 一个乘积 $AB$ 意味着“先做 $B$,再做 $A$”。不移动的点/线是不变的(invariant)。
For a transformation matrix, the determinant gives the: · 对一个变换矩阵,行列式给出:
The determinant is the factor by which areas are scaled by the transformation. · 行列式是变换缩放面积的因子。
For matrices A and B, the product AB means "do B first, then A". · 对矩阵 A 和 B,乘积 AB 意味着“先做 B,再做 A”。
Matrix multiplication is applied right to left: AB(x) = A(B(x)), so B acts first. · 矩阵乘法从右到左应用:AB(x) = A(B(x)),所以 B 先作用。
Common transformations
| Matrix | Transformation |
|---|---|
| $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ | Reflection in $y$-axis |
| $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ | Rotation $90°$ anticlockwise |
| $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ | Enlargement 放大 scale factor $k$ |
- Know matrix addition, the zero matrix, the identity (or unit) matrix, and invariant points and invariant lines of a transformation.
常见的变换
| 矩阵 | 变换 |
|---|---|
| $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ | 在 $y$ 轴上的反射 |
| $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ | 逆时针旋转 $90°$ |
| $\begin{pmatrix} k & 0 \\ 0 & k \end{pmatrix}$ | 比例因子 $k$ 的放大 |
- 掌握矩阵加法(matrix addition)、零矩阵(zero matrix)、单位矩阵(identity (or unit) matrix),以及变换的不变点(invariant points)和不变直线(invariant lines)。
You've got it
- $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$; invertible when $\det \neq 0$
- $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
- a matrix is a transformation; $\det$ = area scale factor; $AB$ = "do $B$, then $A$"
你掌握了
- $\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc$;$\det \neq 0$ 时可逆
- $A^{-1} = \dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
- 一个矩阵是一个变换;$\det$ = 面积比例因子;$AB$ = “先做 $B$,再做 $A$”