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Mathematics差分差分，又名差分函数或差分运算，一般是指有限差分（英语：Finite difference），是数学中的一个概念，将原函数 f ( x ) {\displaystyle f(x)} 映射到 f ( x + a ) − f ( x + b ) {\displaystyle f(x+a)-f(x+b)} 。差分运算，相应于微分运算，是微积分中重要的一个概念。
A finite difference is a mathematical expression of the form f(x + b) − f(x + a). Finite differences (or the associated difference quotients) are often used as approximations of derivatives, such as in numerical differentiation. The difference operator, commonly denoted Δ {\displaystyle \Delta } (uppercase Delta), is the operator that maps a function f to the function Δ [ f ] {\displaystyle \Delta [f]} defined by Δ [ f ] ( x ) = f ( x + 1 ) − f ( x ) . {\displaystyle \Delta [f](x)=f(x+1)-f(x).} A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 傅里叶变换 （法语：Transformation de Fourier，英语：Fourier transform，缩写：FT）是一种线性变换，通常定义为一种积分变换。其基本思想是一个函数可以用（可数或不可数，可数的情况对应于傅里叶级数）无穷多个周期函数的线性组合来逼近，从而这些组合系数在保有原函数的几乎全部信息的同时，还直接地反映了该函数的“频域特征”。因其基本思想首先由法国学者约瑟夫·傅里叶系统地提出，所以以其名字来命名以示纪念。在现代数学理论中，傅里叶积分变换可以得到各种推广，并在分析学中有广泛应用，构成了调和分析这一数学领域。 经过傅里叶变换生成的函数 f ^ {\displaystyle {\hat {f}}} 称作原函数 f {\displaystyle f} 的傅里叶变换，应用意义上称作频谱。在特定情况下，傅里叶变换是可逆的，即将 f ^ {\displaystyle {\hat {f}}} 通过逆变换可以得到其原函数 f {\displaystyle f} 。通常情况下， f {\displaystyle f} 是一个实函数，而 f ^ {\displaystyle {\hat {f}}} 则是一个复数值函数，其函数值作为复数可同时表示振幅和相位。
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both the mathematical operation and to this complex-valued function. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion).
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在数学中，傅里叶级数（英语：Fourier series，/ˈfʊrieɪ, -iər/）是把类似波的函数表示成简单谐波的方式。更正式地说，对于满足狄利克雷定理的周期函数，其傅里叶级数是由一组正弦与余弦函数的加权和表示的方法。傅里叶级数与用来找出无周期函数的频率信息的傅里叶变换有密切的关系。 傅里叶级数是傅里叶分析的一个研究分支，也是采样定理原始证明的核心。傅里叶级数在数论、组合数学、信号处理、概率论、统计学、密码学、声学、光学等领域都有着广泛的应用。
A Fourier series () is a series expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in Fourier series § Definition.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在数学中，函数空间上定义的线性算子 A {\displaystyle A} 的本征函数（英语：Eigenfunction，又称固有函数）就是对该空间中任意一个非零函数 f {\displaystyle f} 进行变换仍然是函数 f {\displaystyle f} 或者其标量倍数的函数。更加精确的描述就是 A f = λ f {\displaystyle {\mathcal {A}}f=\lambda f} 其中 λ 是标量，它是对应的特征值。另外特征值微分的解受到 f {\displaystyle f} 边界条件的限制。当考虑限制条件的时候，只有特定的特征值 λ = λ n {\displaystyle \lambda =\lambda _{n}} （ n = 1 , 2 , 3 , . . .
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f {\displaystyle f} in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue. As an equation, this condition can be written as D f = λ f {\displaystyle Df=\lambda f} for some scalar eigenvalue λ . {\displaystyle \lambda .} The solutions to this equation may also be subject to boundary conditions that limit the allowable eigenvalues and eigenfunctions. An eigenfunction is a type of eigenvector.
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View content license ↗ 在微分方程中，边值问题是一个微分方程和一组称之为边界条件的约束条件。边值问题的解通常是符合约束条件的微分方程的解。 物理学中经常遇到边值问题，例如波动方程等。许多重要的边值问题属于Sturm-Liouville问题。这类问题的分析会和微分算子的本征函数有关。 在实际应用中，边值问题应当是适定的（即：存在解，解唯一且解会随着初始值连续地变化）。许多偏微分方程领域的理论提出是为要证明科学及工程应用的许多边值问题都是适定问题。 最早研究的边值问题是狄利克雷问题，是要找出调和函数，也就是拉普拉斯方程的解，后来是用狄利克雷原理找到相关的解。
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input.
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View content license ↗ 微分方程（英语：Differential equation，DE）是一种数学方程，用来描述某一类函数与其导数之间的关系。微分方程的解是一个满足方程的函数，通常微分方程的解并不唯一，经常要给定恰当的初始条件或边界条件才能确定。而在初等数学的代数方程里，其解是常数。 微分方程的应用十分广泛，可以解决许多与导数有关的问题。物理学中许多涉及变力的运动学、动力学问题，如空气阻力为速度函数的自由落体运动等问题，很多可以用微分方程求解。此外，微分方程在化学、工程学、经济学和数理生物学等领域都有应用。 数学领域对微分方程的研究着重在几个不同的面向，但大多数都是关心微分方程的解。只有少数简单的微分方程可以求得解析解。不过即使没有找到其解析解，仍然可以确认其解的部分性质。在无法求得解析解时，可以利用数值分析的方式，利用电脑来找到其数值解。 动力系统理论强调对于微分方程系统的量化分析，而许多数值方法可以计算微分方程的数值解，且有一定的准确度。
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common in mathematical models and scientific laws; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. The study of differential equations consists mainly of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.
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View content license ↗ MATLAB（Matrix Laboratory，矩阵实验室）是由美国The MathWorks公司出品的商业数学软件。MATLAB是一种用于算法开发、数据可视化、数据分析以及数值计算的高级技术计算语言和交互式环境。除矩阵运算、绘制函数/数据图像等常用功能外，MATLAB还可用来创建用户界面，以及调用其它语言（包括C、C++、Java、Python、FORTRAN）编写的程序。 MATLAB主要用于数值运算，但利用为数众多的附加工具箱，它也适合不同领域的应用，例如控制系统设计与分析、影像处理、深度学习、信号处理与通讯、金融建模和分析等。另外还有配套软件包Simulink提供可视化开发环境，常用于系统模拟、动态/嵌入式系统开发等方面。 在R2017b后的MATLAB版本更新发布了深度学习的工具，使其能够可视化的快速建立AI模型，并透过各种转码器，部属于嵌入式硬件之中。 截至2020年，MATLAB在全球拥有超过400万用户。MATLAB用户来自工程、科学和经济学领域。
MATLAB (Matrix Laboratory) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. As of 2020, MATLAB has more than four million users worldwide. They come from various backgrounds of engineering, science, and economics. As of 2025, more than 6,500 global colleges and universities use MATLAB to support instruction and research.
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View content license ↗ 在数学分支线性代数之中，向量空间中一个向量集合的线性生成空间（linear span，也称为线性包 linear hull），是所有包含这个集合的线性子空间的交集，从而一个向量集合的线性生成空间也是一个向量空间。
In mathematics, the linear span (also called the linear hull or just span) of a set S {\displaystyle S} of elements of a vector space V {\displaystyle V} is the smallest linear subspace of V {\displaystyle V} that contains S . {\displaystyle S.} It is the set of all finite linear combinations of the elements of S, and the intersection of all linear subspaces that contain S . {\displaystyle S.} It is often denoted span(S) or ⟨ S ⟩ . {\displaystyle \langle S\rangle .} For example, in geometry, two linearly independent vectors span a plane. To express that a vector space V is a linear span of a subset S, one commonly uses one of the following phrases: S spans V; S is a spanning set of V; V is spanned or generated by S; S is a generator set or a generating set of V.
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View content license ↗ 线性子空间（或向量子空间）在线性代数和相关的数学领域中是重要的。在没有混淆于其他子空间的时候通常简称为“子空间”。
In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.
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View content license ↗ 在数学里，矩阵加法一般是指两个矩阵把其相对应元素加在一起的运算。但有另一运算也可以认为是一种矩阵的加法。
In mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together. For a vector, v → {\displaystyle {\vec {v}}\!} , adding two matrices would have the geometric effect of applying each matrix transformation separately onto v → {\displaystyle {\vec {v}}\!} , then adding the transformed vectors.
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View content license ↗ 矩阵范数（matrix norm）亦译矩阵模是数学中矩阵论、线性代数、泛函分析等领域中常见的基本概念，是将一定的矩阵空间建立为赋范向量空间时为矩阵装备的范数。应用中常将有限维赋范向量空间之间的映射以矩阵的形式表现，这时映射空间上装备的范数也可以通过矩阵范数的形式表达。
In mathematics, a norm is a function from a vector space to non-negative real numbers that satisfies certain axioms. A matrix norm is a norm defined on a vector space of matrices. As for every norm, a matrix norm defines a distance, the distance between two matrices being the norm of their difference. The specificity of matrix norms is that they may often be defined either directly as for every norm, or by mean of their properties as linear operators. There are many specific matrix norms. Most arise from the following three perspectives, though different perspectives may sometimes yield the same norm. Consider the matrix as a linear operator; then a matrix norm may describe how much the operator can stretch vectors. Such matrix norms induced by vector norms are called operator norms. Consider the matrix as a rectangular array of numbers; then a matrix norm may be defined as a function of the entries of the matrix. Such matrix norms are sometimes called "entry-wise" norms. The singular value decomposition is useful in analyzing matrices.
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View content license ↗ Mathcad是一种交互式数值计算系统。当输入一个数学公式、方程组、矩阵等，计算机将直接给出计算结果，而无须去考虑中间计算过程。因而Mathcad在航空、国防、消费品设计等科学和工程领域中承担着复杂的数学计算，图形显示和文档处理，是工程技术人员常用的工具。Mathcad有五个扩展库，分别是求解与优化，数据分析，信号处理，图像处理和小波分析。Mathcad是美国PTC公司的产品。
Mathcad is computer software for the verification, validation, documentation and re-use of mathematical calculations in engineering and science, notably mechanical, chemical, electrical, and civil engineering. Released in 1986 for MS-DOS, it introduced live editing (WYSIWYG) of typeset mathematical notation in an interactive notebook, combined with automatic computations. It was originally developed by Mathsoft, and since 2006 has been a product of Parametric Technology Corporation.
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View content license ↗ 在数学中，矩阵微积分是多元微积分的一种特殊表达，尤其是在矩阵空间上进行讨论的时候。它把单个函数对多个变量或者多元函数对单个变量的偏导数写成向量和矩阵的形式，使其可以被当成一个整体被处理。这使得要在多元函数寻找最大或最小值，又或是要为微分方程系统寻解的过程大幅简化。这里我们主要使用统计学和工程学中的惯用记法，而张量下标记法更常用于物理学中。
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices. It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities. This greatly simplifies operations such as finding the maximum or minimum of a multivariate function and solving systems of differential equations. The notation used here is commonly used in statistics and engineering, while the tensor index notation is preferred in physics. Two competing notational conventions split the field of matrix calculus into two separate groups. The two groups can be distinguished by whether they write the derivative of a scalar with respect to a vector as a column vector or a row vector. Both of these conventions are possible even when the common assumption is made that vectors should be treated as column vectors when combined with matrices (rather than row vectors).
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View content license ↗ 在线性代数，特别是二次型理论中，常常用到矩阵间的合同关系。两个矩阵 A {\displaystyle A} 和 B {\displaystyle B} 是合同的，如果有同数域上的可逆矩阵 P {\displaystyle P} ，使得 A = P T B P {\displaystyle A=P^{\mathrm {T} }BP\,} 。 其中的 P T {\displaystyle P^{\mathrm {T} }} 表示矩阵 P {\displaystyle P} 的转置矩阵。 对于二次型的矩阵表示来说，做一次非退化的线性替换相当于将二次型的矩阵变为一个与其合同的矩阵。 在有限维线性空间中同一双线性函数在不同基下的度量矩阵是合同的。
In mathematics, two square matrices A {\displaystyle A} and B {\displaystyle B} over a field are called congruent if there exists an invertible matrix P {\displaystyle P} over the same field such that P T A P = B {\displaystyle P^{\mathsf {T}}AP=B} Matrix congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases. Halmos defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose, but this definition has not been adopted by most other authors.
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View content license ↗ 矩阵差分方程是一种差分方程，其中某时刻的变量向量（或矩阵）与之前时刻的值通过矩阵相关。方程的阶是变量向量任意两个指示值之间的最大时差。例如 x t = A x t − 1 + B x t − 2 {\displaystyle \mathbf {x} _{t}=\mathbf {Ax} _{t-1}+\mathbf {Bx} _{t-2}} 是二阶矩阵差分方程，其中x是n × 1变量向量，A、B是n × n矩阵。该方程齐次，因为方程末尾没有常数项向量。同一个方程也可写成 x t + 2 = A x t + 1 + B x t {\displaystyle \mathbf {x} _{t+2}=\mathbf {Ax} _{t+1}+\mathbf {Bx} _{t}}…
A matrix difference equation is a difference equation in which the value of a vector (or sometimes, a matrix) of variables at one point in time is related to its own value at one or more previous points in time, using matrices. The order of the equation is the maximum time gap between any two indicated values of the variable vector. For example, x t = A x t − 1 + B x t − 2 {\displaystyle \mathbf {x} _{t}=\mathbf {Ax} _{t-1}+\mathbf {Bx} _{t-2}} is an example of a second-order matrix difference equation, in which x is an n × 1 vector of variables and A and B are n × n matrices. This equation is homogeneous because there is no vector constant term added to the end of the equation.
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View content license ↗ 矩阵分析（英语：matrix analysis） 是一门研究矩阵及其代数性质的学科。这门学科研究的内容包括矩阵的运算（加法、矩阵乘法等）、矩阵函数、矩阵的特征值（特征值分解）等。
In mathematics, particularly in linear algebra and applications, matrix analysis is the study of matrices and their algebraic properties. Some particular topics out of many include; operations defined on matrices (such as matrix addition, matrix multiplication and operations derived from these), functions of matrices (such as matrix exponentiation and matrix logarithm, and even sines and cosines etc. of matrices), and the eigenvalues of matrices (eigendecomposition of a matrix, eigenvalue perturbation theory).
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View content license ↗ LoRA（Low-Rank Adaptation，意为低秩适应）是一种用于大型语言模型及其他深度神经网络的参数高效微调技术。这项技术由微软研究人员于2021年提出。相较于传统的全模型微调，LoRA能在极大减少计算资源和可训练参数的前提下，将预训练模型适应至特定任务。
LoRA (Low-Rank Adaptation) is a parameter-efficient fine-tuning technique for large language models and other deep neural networks. Introduced in 2021 by researchers at Microsoft, LoRA enables adaptation of pre-trained models to specific tasks while requiring significantly fewer computational resources and trainable parameters than traditional full model fine-tuning.
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View content license ↗ 在多重线性代数中，多重线性形式是 f : V N → K {\displaystyle f:V^{N}\to K} 类型的映射，这里的 V 是在域 K 上的向量空间，它分别在其 N 个变量的每个之上是线性的。 单词“形式”通常称呼从向量空间到它的底层域的映射，对在其所有参数上都是线性的一般映射使用更一般的术语多重线性映射。 对于 N = 2，就是说只有两个变量，称 f 为双线性形式。 一种重要的多重线性形式是“交替多重线性形式”，它有在交换两个参数的时候改变其正负号的额外性质。当 K 的特征不是 2 的时候，这等价于说 f ( … , x , … , x , … ) = 0 {\displaystyle f(\dots ,x,\dots ,x,\dots )=0} , 就是说在提供同一个参数两次的时候这个形式变为零。(特征 2 的异常情况需要更加小心)。这些特殊情况有行列式形式和微分形式。
In abstract algebra and multilinear algebra, a multilinear form on a vector space V {\displaystyle V} over a field K {\displaystyle K} is a map f : V k → K {\displaystyle f\colon V^{k}\to K} that is separately K {\displaystyle K} -linear in each of its k {\displaystyle k} arguments. More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 范数（英语：Norm），是具有“长度”概念的函数。在线性代数、泛函分析及相关的数学领域，是一个函数，其为向量空间内的所有向量赋予非零的正长度或大小。另一方面，半范数（英语：seminorm）可以为非零的向量赋予零长度。 举一个简单的例子，一个二维度的欧几里得空间 R 2 {\displaystyle \mathbb {R} ^{2}} 就有欧氏范数。在这个向量空间的元素（譬如： ( 3 , 7 ) {\displaystyle (3,7)} ）常常在笛卡尔坐标系统被画成一个从原点出发的箭号。每一个向量的欧氏范数就是箭号的长度。 拥有范数的向量空间就是赋范向量空间。同样，拥有半范数的向量空间就是赋半范向量空间。
In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude or length of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". It can also refer to a norm that can take infinite values or to certain functions parametrised by a directed set.
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View content license ↗ 在线性代数和泛函分析的数学领域中，内积空间 V 的子空间 W 的正交补（英语：orthogonal complement） W ⊥ {\displaystyle W^{\bot }} 是正交于 W 中所有向量的所有 V 中向量的集合，也就是 W ⊥ = { x ∈ V : ∀ y ∈ W , ⟨ x , y ⟩ = 0 } . {\displaystyle W^{\bot }=\left\{\,x\in V:\forall y\in W,\langle x,y\rangle =0\,\right\}.\,} 正交补总是闭合在度量拓扑下。在希尔伯特空间中，W 的正交补的正交补是 W 的闭包，就是说 W ⊥ ⊥ = W ¯ .
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace W {\displaystyle W} of a vector space V {\displaystyle V} equipped with a bilinear form B {\displaystyle B} is the set W ⊥ {\displaystyle W^{\perp }} of all vectors in V {\displaystyle V} that are orthogonal to every vector in W {\displaystyle W} . Informally, it is called the perp, short for perpendicular complement. It is a subspace of V {\displaystyle V} .
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View content license ↗ 卦限是笛卡儿坐标系中，象限在三维空间的对应术语，用于空间解析几何的坐标系统。空间直角坐标系用于确定空间的任意一点的位置。先在指定空间内的任意一点取定并标记点 O，作为原点。经过点 O，画出三条互相垂直的直线，把它们分别标记作 x 轴、y 轴和z 轴。用右手定则规定各轴线的正方向。每二条轴确定出一个平面，作为坐标平面。由 x 轴和 y 轴确定的坐标平面称作 xy 平面；x 轴、z 轴确定 xz 平面；最后一对，y、z 二轴确定 yz 平面。按照传统，将 xy 平面配置在水平面上，z 轴置于铅直位置，而 xz、yz 二平面在图上垂直标示。这三个坐标平面将空间分为八个部分，这便是空间直角坐标系的8个卦限。 八个卦限在几何图中通常以罗马数字“I、II、III、IV、V、VI、VII、VIII”标示。较为普遍的卦限数序均以 x 轴正半轴、y 轴正半轴和 z 轴正半轴确定的卦限为“第一卦限”，罗马数字标记为“I”。第二、三、四卦限的数序类似笛卡尔坐标系中象限的数序。在 xy 平面上向逆时针方向增加数序。而后第五至八卦限在 xy 平面下同样以逆时针方向标记。 因卦限相对象限较为罕见，世界各地的数学家乃至不同时代的数学印刷物都曾使用过不同的数序来标记各个卦限，所以为了避免混淆，可以采用另一种标记卦限的方式。直接地，明确指出某卦限范围内包含的 x、y、z 坐标的正负，来标记那个卦限。如图1中的第一卦限（I）标作“(+,+,+)”；第四卦限（IV）标作“(+,-,+)”；第七卦限（VII）标作“(-,-,-)”。
In geometry, an orthant or hyperoctant is the analogue in n-dimensional Euclidean space of a quadrant in the plane or an octant in three dimensions. In general an orthant in n-dimensions can be considered the intersection of n mutually orthogonal half-spaces. By independent selections of half-space signs, there are 2n orthants in n-dimensional space. More specifically, a closed orthant in Rn is a subset defined by constraining each Cartesian coordinate to be nonnegative or nonpositive. Such a subset is defined by a system of inequalities: ε1x1 ≥ 0 ε2x2 ≥ 0 · · · εnxn ≥ 0, where each εi is +1 or −1. Similarly, an open orthant in Rn is a subset defined by a system of strict inequalities ε1x1 > 0 ε2x2 > 0 · · · εnxn > 0, where each εi is +1 or −1. By dimension: In one dimension, an orthant is a ray. In two dimensions, an orthant is a quadrant. In three dimensions, an orthant is an octant. John Conway and Neil Sloane defined the term n-orthoplex from orthant complex as a regular polytope in n-dimensions with 2n simplex facets, one per orthant.
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View content license ↗ 数学中，实向量空间的一个定向（Orientation）是对哪些有序基是“正”定向以及哪些是“负”定向的一个选取。在三维欧几里得空间中，两个可能的基本定向分别称为右手系与左手系。但是定向的选取与基的手征性是独立的（尽管右手基典型地选为正定向，但它们也可规定为负定向）。
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary. A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected is called unoriented. In mathematics, orientability is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra over the real numbers, the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple displacement.
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View content license ↗ 数学中，牛顿恒等式（英语：Newton's identities）描述了幂和对称多项式和初等对称多项式此两种对称多项式之间的关系。 牛顿在不知道阿尔伯特‧吉拉德先前的成果下，于约1666年发现这些恒等式。这些恒等式目前已被应用在许多数学领域，如伽罗瓦理论、不变量理论、群论、组合学，也被进一步应用于数学之外，如广义相对论。
In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity.
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View content license ↗ 非线性特征值问题是特征值, 非线性依赖于特征值的方程的特征值问题的推广. 具体来说, 非线性特征值问题指的是具以下形式的方程: A ( λ ) x = 0 , {\displaystyle A(\lambda )\mathbf {x} =0,\,} 其中 x 是向量(非线性"特征向量"), A 是 λ {\displaystyle \lambda } (非线性"特征根")的函数矩阵.(更一般的, A ( λ ) {\displaystyle A(\lambda )} 可以是一个线性映射, 但最常用的是有限维矩阵, 通常为方阵.) 通常要求 A 为 λ {\displaystyle \lambda } (在某个定义域内)的全纯函数. 例如, 特征值问题 B v = λ v {\displaystyle B\mathbf {v} =\lambda \mathbf {v} } , 其中 B 为方阵, 对应于 A ( λ ) = B − λ I {\displaystyle A(\lambda )=B-\lambda I} 的特征值问题, 其中 I 是单位矩阵. 常见的情况是多项式特征值问题, 其中 A 为多项式矩阵.
In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form M ( λ ) x = 0 , {\displaystyle M(\lambda )x=0,} where x ≠ 0 {\displaystyle x\neq 0} is a vector, and M {\displaystyle M} is a matrix-valued function of the number λ {\displaystyle \lambda } . The number λ {\displaystyle \lambda } is known as the (nonlinear) eigenvalue, the vector x {\displaystyle x} as the (nonlinear) eigenvector, and ( λ , x ) {\displaystyle (\lambda ,x)} as the eigenpair.
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View content license ↗ 在线性代数里，假若，内积空间的两个向量是互相正交的，并且，两个向量的范数都是 1 ，则称这两个向量互相具有正交规范性，又译单范正交性，正交归一性。假若，一组向量全都是互相正交规范的，则称这组向量为正交规范集。假若，这正交规范集形成了一个基，则称这集合为正交规范基。
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the two vectors are perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
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View content license ↗ 正交基（英语：orthogonal basis）在线性代数中是指内积空间的一些特定基底，其元素（称作基向量）两两正交。若一组正交基的元素均为单位向量（其范数皆为1），则称这组正交基为标准正交基（英语：orthonormal basis，亦称规范正交基）。 无论在有限维还是无限维空间中，正交基的概念都是很重要的。在无限维希尔伯特空间中，正交基不再是哈默尔基，也就是说不是每个元素都可以写成有限个基中元素的线性组合。因此在无限维空间中，正交基应该被更严格地定义为由线性无关而且两两正交的元素组成、张成的空间是原空间的一个稠密子空间（而不是整个空间）的集合。 注意，在没有定义内积的空间中，“正交基”一词是没有意义的。因此，一个具有正交基的巴拿赫空间，就是一个希尔伯特空间。
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V {\displaystyle V} with finite dimension is a basis for V {\displaystyle V} whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for R n {\displaystyle \mathbb {R} ^{n}} arises in this fashion. An orthonormal basis can be derived from an orthogonal basis via normalization.
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View content license ↗ 线性代数中的正交化指的是：从内积空间（包括常见的欧几里得空间）中的一组线性无关向量v1，...，vk出发，得到同一个子空间上两两正交的向量组u1，...，uk。 如果还要求正交化后的向量都是单位向量，那么称为标准正交化。 一般在数学分析中采用格拉姆-施密特正交化作正交化的计算。在编程计算时，格拉姆-施密特正交化的数值稳定性不高，所以常用更稳定的豪斯霍尔德变换代替。另外，相对于豪斯霍尔德变换在最后直接生成所有的向量，格拉姆-施密特方法在第i步产生第i个向量，因此后者可用迭代法编写。对于含有零元素较多的向量组（例如稀疏矩阵的QR分解），还会采用吉文斯旋转。
In linear algebra, orthogonalization is the process of finding a set of orthogonal vectors that span a particular subspace. Formally, starting with a linearly independent set of vectors {v1, ... , vk} in an inner product space (most commonly the Euclidean space Rn), orthogonalization results in a set of orthogonal vectors {u1, ... , uk} that generate the same subspace as the vectors v1, ... , vk. Every vector in the new set is orthogonal to every other vector in the new set; and the new set and the old set have the same linear span. In addition, if we want the resulting vectors to all be unit vectors, then we normalize each vector and the procedure is called orthonormalization. Orthogonalization is also possible with respect to any symmetric bilinear form (not necessarily an inner product, not necessarily over real numbers), but standard algorithms may encounter division by zero in this more general setting.
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View content license ↗ 在线性代数中，积和式（英语：permanent）是一个由方块矩阵 A {\displaystyle A} 计算得到的标量，记作 perm ( A ) {\displaystyle \operatorname {perm} (A)} 。积和式的定义与行列式类似，只是在求和时不添加正负号。当矩阵 A {\displaystyle A} 包含若干变量时，积和式也可以看作是一个关于这些变量的多项式。积和式在计算机科学，特别是计算复杂性理论中有重要的地位，因为理论上的一个重要难题——计算一个二分图（英语：bipartite graph）上完美匹配（英语：perfect matching）的数目——等价于求某个矩阵的积和式。
In linear algebra, the permanent of a square matrix is a function of the matrix similar to the determinant. The permanent, as well as the determinant, is a polynomial in the entries of the matrix. Both are special cases of a more general function of a matrix called the immanant.
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View content license ↗ 赝标量（英语：pseudoscalar）为类似标量的数量，但在空间反演、瑕旋转时会多出负号，标量则不会。 赝向量与向量的内积会是赝标量。赝标量的一个典型例子为三重积。设空间中有三向量A、B、C，彼此线性无关；A与B的叉积 A × B {\displaystyle \mathbf {A} \times \mathbf {B} } 为一赝向量，此叉积再与C做内积可得三重积 ( A × B ) ⋅ C {\displaystyle (\mathbf {A} \times \mathbf {B} )\cdot \mathbf {C} } ，即A、B与C所构成的平行六面体体积。赝标量与向量的乘积会产生赝向量；赝标量与张量的乘积会产生赝张量。
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. A pseudoscalar, when multiplied by an ordinary vector, becomes a pseudovector (or axial vector); a similar construction creates the pseudotensor. A pseudoscalar also results from any scalar product between a pseudovector and an ordinary vector. The prototypical example of a pseudoscalar is the scalar triple product, which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where the latter is a pseudovector.
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View content license ↗ 在线性代数和泛函分析中，投影是从向量空间映射到自身的一种线性变换 P {\displaystyle P} ，满足 P 2 = P {\displaystyle P^{2}=P} ,也就是说，当 P {\displaystyle P} 两次作用于某个值，与作用一次得到的结果相同（幂等）。是日常生活中“平行投影”概念的形式化和一般化。同现实中阳光将事物投影到地面上一样，投影变换将整个向量空间映射到它的其中一个子空间，并且在这个子空间中是恒等变换。
In linear algebra and functional analysis, a projection is a linear transformation P {\displaystyle P} from a vector space to itself (an endomorphism) such that P ∘ P = P {\displaystyle P\circ P=P} . That is, whenever P {\displaystyle P} is applied twice to any vector, it gives the same result as if it were applied once (i.e. P {\displaystyle P} is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.
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