奇异值分解（SVD）把任意矩阵分解成什么形式，它有什么用途？
Into what form does the singular value decomposition factor any matrix, and what is it used for?
奇异值分解把任意矩阵写成三个矩阵的乘积：一个正交矩阵、一个含非负奇异值的对角矩阵，以及另一个正交矩阵的转置。它对任意矩阵都存在，可用于数据压缩、伪逆与降维。戈卢布与范洛恩的《矩阵计算》是这一算法的权威参考。
The singular value decomposition factors any matrix into a product of an orthogonal matrix, a diagonal matrix of non-negative singular values, and the transpose of another orthogonal matrix. It exists for every matrix and is used in compression, pseudoinverses, and dimensionality reduction. Golub & Van Loan's Matrix Computations is the authoritative reference.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-11-02
来源 · Source
Golub 与 Van Loan《Matrix Computations》；Strang《Linear Algebra and Its Applications》
Golub & Van Loan, Matrix Computations; Strang, Linear Algebra and Its Applications
独立核验依据 · Independent verification: 核验依据 1 OPENLIBRARY 标识 /works/OL3285496W：Linear algebra and its
#奇异值分解 #正交矩阵 #降维
谱定理是什么？
What is the spectral theorem?
谱定理说实对称矩阵（或更一般的自伴算子）可被一组正交特征向量对角化，特征值全为实数。它是主成分分析、振动模态和量子力学的数学基础。
The spectral theorem says a real symmetric (self-adjoint) operator is orthogonally diagonalizable with real eigenvalues; the basis of PCA and quantum mechanics.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#线性代数 #谱定理
摩尔-彭罗斯伪逆是什么？
What is the Moore–Penrose pseudoinverse?
当矩阵不是方阵或不满秩时，伪逆 A⁺ 给出“最佳”的最小二乘解 x = A⁺b。它把可逆矩阵的逆推广到所有矩阵，广泛用于数据拟合与机器学习。
The Moore–Penrose pseudoinverse A⁺ gives the best least-squares solution x=A⁺b even for non-square or rank-deficient matrices; a generalization of inversion.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#线性代数 #伪逆
格拉姆-施密特过程如何把一组线性无关向量正交化？
How does the Gram-Schmidt process orthogonalize a set of linearly independent vectors?
格拉姆-施密特过程逐个取出向量，减去它在已构造正交基上各方向的投影，得到彼此正交的新向量。它把任意线性无关组转化为正交甚至标准正交基，是QR分解与数值计算的基础。斯特朗与切尼、金凯德的教材都给出标准算法。
The Gram-Schmidt process takes vectors one by one and subtracts their projections onto the already constructed orthogonal basis, yielding mutually orthogonal vectors. It turns any linearly independent set into an orthogonal or orthonormal basis and underlies QR factorization and numerical work. Strang and Cheney & Kincaid give the standard algorithm.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
Strang《Linear Algebra》；Cheney 与 Kincaid《Numerical Mathematics》
Strang, Linear Algebra; Cheney & Kincaid, Numerical Mathematics
#格拉姆-施密特 #正交化 #标准正交基
谷歌的PageRank是怎样把亿万网页之间的链接关系压缩成一个矩阵方程的？“最重要的页面”为什么恰好等于某个特定矩阵的主特征向量？
How does Google's PageRank compress the link relations among billions of webpages into a single matrix equation, and why is the most important page exactly captured by the dominant eigenvector of a particular matrix?
把每个页面的出链权重按目标均分成列随机矩阵，再混入允许随机跳转的阻尼项，得到元素全为正的Google矩阵；排名向量满足Gr=r，也就是特征值1对应的特征向量。Perron–Frobenius定理保证这个主特征值单重且特征向量严格为正，因此排名存在且唯一。实际计算只需幂法迭代几十次即可收敛，这成为线性代数支撑现代搜索的经典案例。
Each page splits its outgoing-link weight evenly across its targets to form a column-stochastic matrix, which is blended with a uniform random-jump damping term into a strictly positive Google matrix; the ranking vector satisfies Gr=r, meaning it is an eigenvector for eigenvalue 1. The Perron–Frobenius theorem guarantees that this dominant eigenvalue is simple with a strictly positive eigenvector, so the ranking exists and is unique. In practice a few dozen power-method iterations suffice, making this the canonical example of linear algebra powering modern search.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录含书名，但缺版次 / 出版社 / 年份中的一部分，需编辑补全后才能做独立比对。
劳伦斯·佩奇、谢尔盖·布林等著《PageRank引用排序：为Web带来秩序》，斯坦福大学InfoLab技术报告（1999）
Lawrence Page, Sergey Brin, Rajeev Motwani, Terry Winograd, 'The PageRank Citation Ranking: Bringing Order to the Web', Stanford InfoLab technical report (1999)
#特征向量排序 #随机矩阵
矩阵对角化是什么？
What is matrix diagonalization?
若方阵 A 有完备的特征向量基，则可写成 A = PDP⁻¹，其中 D 是对角阵、P 由特征向量组成。对角化让高次幂 Aⁿ = PDⁿP⁻¹ 与求解微分方程变得简单。
If A has a full eigenbasis, A=PDP⁻¹ with diagonal D; then Aⁿ=PDⁿP⁻¹, greatly simplifying powers and differential equations.
难度 入门 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#线性代数 #对角化
什么是范数？
What is a norm?
范数把一个向量“长度化”为一个非负实数，满足非负性、齐次性和三角不等式。常见如 L1（绝对值和）、L2（欧几里得长度）、L∞（最大分量）。
A norm assigns each vector a nonnegative 'length' satisfying positivity, homogeneity and the triangle inequality; examples are L1, L2 and L∞ norms.
难度 入门 · 适合 小学低段(6-8)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#线性代数 #范数
什么是正交投影？
What is an orthogonal projection?
正交投影把向量“垂直地”投到某个子空间上，使误差向量与子空间垂直。最小二乘拟合本质上就是求数据点在“模型子空间”上的正交投影。
An orthogonal projection maps a vector perpendicularly onto a subspace so the error is orthogonal to it; least-squares fitting is exactly such a projection.
难度 入门 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#线性代数 #正交投影
秩-零度定理是什么？
What is the rank–nullity theorem?
秩-零度定理说：线性变换的秩（像的维数）与其零度（核的维数）之和等于定义域维数，即 rank + nullity = n。它揭示了“信息保留”与“信息丢失”的此消彼长。
Rank–nullity: rank (dimension of image) plus nullity (dimension of kernel) equals the domain dimension n; it balances information kept vs. lost.
难度 入门 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-08-23 · 2 条独立证据
来源 · Source
Courant & Robbins,《What Is Mathematics?》Oxford University Press,1996 第二版;中译《什么是数学》
R. Courant and H. Robbins, What Is Mathematics?, Oxford University Press, 1996; Chinese ed.
独立核验依据 · Independent verification: 核验依据 2 OPENLIBRARY 标识 /works/OL811195W：What is mathematics?（按标题
#线性代数 #秩零度