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Mathematics连通性在数学中，连通性用于指代各种属性，在某种意义上意味着“整体”。当一个数学对象具有这样的属性时，我们说它是连通的；否则连接断开。当一个断开连接的对象可以自然地分割成连接的部分时，每个部分通常称为一个组件（或连接的组件）。
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be split naturally into connected pieces, each piece is usually called a component (or connected component).
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数值分析中，数值方法是一种旨在解决数值问题的数学工具。用编程语言实现具有适当收敛性检查的数值方法称为数值算法。
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在概率论与统计学中，协方差（英语：Covariance）用于衡量随机变量间的相关程度。
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. The sign of the covariance shows the tendency in the linear relationship between the variables. Covariance is positive when variables tend to show similar behavior and negative when variables tend to show opposite behavior. The magnitude of the covariance is the geometric mean of the variances that are shared for the two random variables, where a larger magnitude means two variables more strongly depend on each other. Covariance has units of measurement, and the magnitude of the covariance is affected by said units. This means changing the units (e.g., from meters to millimeters) changes the covariance value proportionally, making it difficult to assess the strength of the relationship from the covariance alone.
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View content license ↗ 再现性与可重复性和可重复性密切相关，是支撑科学方法的主要原则。研究结果的可重复性意味着，当研究被重复时，通过实验或观察性研究或数据集统计分析获得的结果应该再次获得高度的可靠性。复制有不同类型，但复制研究通常涉及使用相同方法的不同研究人员。只有在一次或多次成功复制之后，结果才能被视为科学知识。
Reproducibility, closely related to replicability and repeatability, is a major principle underpinning the scientific method. For the findings of a study to be reproducible means that results obtained by an experiment or an observational study or in a statistical analysis of a data set should be achieved again with a high degree of reliability when the study is replicated. There are different kinds of replication but typically replication studies involve different researchers using the same methodology. Only after one or several such successful replications should a result be recognized as scientific knowledge.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在线性代数和相关数学领域中，向量空间中的平衡集、圆集或圆盘（在具有绝对值函数 | ⋅ | {\displaystyle |\cdot |} 的域 K {\displaystyle \mathbb {K} } 上）是一个集合 S {\displaystyle S} ，使得 S ⊆ S {\displaystyle aS\subseteq S} 对于所有标量 a {\displaystyle a} 满足 |一个 | ≤ 1. {\displaystyle |a|\leq 1.} 集合 S {\displaystyle S} 的平衡外壳或平衡包络是包含 S 的最小平衡集合。
In linear algebra and related areas of mathematics a balanced set, circled set or disk in a vector space (over a field K {\displaystyle \mathbb {K} } with an absolute value function | ⋅ | {\displaystyle |\cdot |} ) is a set S {\displaystyle S} such that a S ⊆ S {\displaystyle aS\subseteq S} for all scalars a {\displaystyle a} satisfying | a | ≤ 1. {\displaystyle |a|\leq 1.} The balanced hull or balanced envelope of a set S {\displaystyle S} is the smallest balanced set containing S .
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在几何中，重心坐标系是通过参考单纯形（平面中的点为三角形，三维空间中的点为四面体等）来指定点位置的坐标系。点的重心坐标可以解释为放置在单纯形顶点处的点质量，使得该点是这些质量的质心（或重心）。这些质量可以为零或负数；当且仅当该点严格位于单纯形内时，它们都是正的。每个点都有重心坐标，它们的总和永远不为零。当且仅当它们成比例时，两个重心坐标元组指定同一点；也就是说，如果一个元组可以通过将另一个元组的元素乘以相同的非零数得到。
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as point masses placed at the vertices of the simplex, such that the point is the center of mass (or barycenter) of these masses. These masses can be zero or negative; they are all positive if and only if the point is strictly inside the simplex. Every point has barycentric coordinates, and their sum is never zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，如果 V 的每个元素都可以以唯一的方式写为 B 元素的有限线性组合，则向量空间 V 的元素集合 B 称为基（复数：基）。该线性组合的系数称为向量相对于 B 的分量或坐标。基的元素称为基向量。等价地，如果集合 B 的元素是线性无关的，并且 V 的每个元素都是 B 元素的线性组合，则该集合 B 是基。换句话说，基是线性无关的生成集。向量空间可以有多个基；然而，所有基都具有相同数量的元素，称为向量空间的维数。本文主要讨论有限维向量空间。然而，许多原理对于无限维向量空间也有效。
In mathematics, a set B of elements of a vector space V is called a basis (pl.: bases) if every element of V can be written in a unique way as a finite linear combination of elements of B. The coefficients of this linear combination are referred to as components or coordinates of the vector with respect to B. The elements of a basis are called basis vectors. Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B. In other words, a basis is a linearly independent spanning set. A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，基函数是函数空间的特定基的元素。函数空间中的每个函数都可以表示为基函数的线性组合。在有限维向量空间中，这种表示是纯代数的，并且仅涉及有限多个基函数，而在无限维设置中，它可能采取无限级数或另一个限制过程的形式，其收敛取决于空间的拓扑。基函数的选择不是唯一的。不同的基可以表示相同的函数空间，但可以具有对特定应用有用的不同属性。例如，单项式对于初等多项式计算很方便，三角函数对于傅里叶分析非常有用，局部支持函数对于数值方法非常有用。
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions. In finite-dimensional vector spaces, this representation is purely algebraic and involves only finitely many basis functions, whereas in infinite-dimensional settings it may take the form of an infinite series or another limiting process whose convergence depends on the topology of the space. The choice of basis functions is not unique. Different bases can represent the same function space, but can have different properties that are useful for particular applications. For example, monomials are convenient for elementary polynomial calculations, trigonometric functions are useful for Fourier analysis, and locally supported functions are useful in numerical methods.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，本迪克森不等式是Ivar Bendixson于1902年导出的矩阵领域的定量结果。该不等式对实矩阵的特征根（特征值）的虚部和实部施加了限制。该不等式的一个特例导致实对称矩阵的特征根始终为实数。
In mathematics, Bendixson's inequality is a quantitative result in the field of matrices derived by Ivar Bendixson in 1902. The inequality puts limits on the imaginary and real parts of characteristic roots (eigenvalues) of real matrices. A special case of this inequality leads to the result that characteristic roots of a real symmetric matrix are always real.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，双对角矩阵是沿主对角线以及上方对角线或下方对角线具有非零条目的带状矩阵。这意味着矩阵中有两条非零对角线。当主对角线上方的对角线具有非零项时，矩阵是上双对角矩阵。当主对角线下方的对角线具有非零条目时，矩阵是下双对角矩阵。
In mathematics, a bidiagonal matrix is a banded matrix with non-zero entries along the main diagonal and either the diagonal above or the diagonal below. This means there are exactly two non-zero diagonals in the matrix. When the diagonal above the main diagonal has the non-zero entries the matrix is upper bidiagonal. When the diagonal below the main diagonal has the non-zero entries the matrix is lower bidiagonal.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在运筹学中，大M方法是一种使用单纯形算法解决线性规划问题的方法。 Big M 方法将单纯形算法扩展到包含“大于”约束的问题。它通过将约束与大负常量相关联来实现这一点，这些负常量不会成为任何最佳解决方案（如果存在）的一部分。
In operations research, the Big M method is a method of solving linear programming problems using the simplex algorithm. The Big M method extends the simplex algorithm to problems that contain "greater-than" constraints. It does so by associating the constraints with large negative constants which would not be part of any optimal solution, if it exists.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，双线性形式是向量空间 V（其元素称为向量）在域 K（其元素称为标量）上的双线性映射 V × V → K。换句话说，双线性形式是一个函数 B : V × V → K，在每个参数中分别是线性的： B(u + v, w) = B(u, w) + B(v, w) 和 B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) 和 B(u, λv) = λB(u, v) R n {\displaystyle \mathbb {R} ^{n}} 是双线性形式的一个例子，它也是内积。非内积的双线性形式的一个示例是四向量积。双线性形式的定义可以扩展到包括环上的模，并用模同态代替线性映射。
In mathematics, a bilinear form is a bilinear map V × V → K on a vector space V (the elements of which are called vectors) over a field K (the elements of which are called scalars). In other words, a bilinear form is a function B : V × V → K that is linear in each argument separately: B(u + v, w) = B(u, w) + B(v, w) and B(λu, v) = λB(u, v) B(u, v + w) = B(u, v) + B(u, w) and B(u, λv) = λB(u, v) The dot product on R n {\displaystyle \mathbb {R} ^{n}} is an example of a bilinear form which is also an inner product. An example of a bilinear form that is not an inner product would be the four-vector product. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在线性代数中，块LU分解是将块矩阵分解为下块三角矩阵L和上块三角矩阵U。这种分解用于数值分析，以降低块矩阵公式的复杂度。
In linear algebra, a Block LU decomposition is a matrix decomposition of a block matrix into a lower block triangular matrix L and an upper block triangular matrix U. This decomposition is used in numerical analysis to reduce the complexity of the block matrix formula.
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View content license ↗ Bra–ket 表示法或 Dirac 表示法是有限维和无限维情况下复向量空间及其对偶空间上线性代数和线性算子的数学表示法。它是专门为简化量子力学中经常出现的计算类型而设计的。现在它在该主题中已普遍使用。括号表示法是由保罗·狄拉克（Paul Dirac）在其1930年牛津大学出版的专着《量子力学原理》中创建的。名称来源于英文单词“bracket”。
Bra–ket notation or Dirac notation is a mathematical notation for linear algebra and linear operators on complex vector spaces together with their dual spaces both in the finite- and infinite-dimensional cases. It is specifically designed to ease the types of calculations that frequently arise in quantum mechanics. It is now of ubiquitous usage in that subject. Bra–ket notation was created by Paul Dirac in his monograph, "The Principles of Quantum Mechanics" published by Oxford University 1930. The name comes from the English word bracket.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，特别是线性代数中，以 James R. Bunch、Christopher P. Nielsen 和 Danny C. Sorensen 命名的 Bunch-Nielsen-Sorensen 公式表示对称矩阵 A {\displaystyle A} 与向量 v {\displaystyle v} 与其自身的外积 v v T {\displaystyle vv^{T}} 之和的特征向量。
In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a symmetric matrix A {\displaystyle A} and the outer product, v v T {\displaystyle vv^{T}} , of vector v {\displaystyle v} with itself.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，规范基是代数结构的基础，在某种意义上是规范的，取决于精确的上下文：在坐标空间中，更一般地在自由模中，它指的是由克罗内克三角洲定义的标准基。在多项式环中，它指的是由单项式 ( X i ) i {\displaystyle (X^{i})_{i}} 给出的标准基。对于有限可拓域，它意味着多项式基。在线性代数中，它指的是 n×n 矩阵 A {\displaystyle A} 的 n 个线性独立广义特征向量的集合，如果该集合完全由乔丹链组成。在表示论中，它指的是Lusztig引入的量子群的基础。
In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta. In a polynomial ring, it refers to its standard basis given by the monomials, ( X i ) i {\displaystyle (X^{i})_{i}} . For finite extension fields, it means the polynomial basis. In linear algebra, it refers to a set of n linearly independent generalized eigenvectors of an n×n matrix A {\displaystyle A} , if the set is composed entirely of Jordan chains. In representation theory, it refers to the basis of the quantum groups introduced by Lusztig.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在几何和线性代数中，笛卡尔张量使用标准正交基以分量的形式表示欧几里得空间中的张量。将张量的分量从一种这样的基础转换为另一种基础是通过正交变换来完成的。最熟悉的坐标系是二维和三维笛卡尔坐标系。笛卡尔张量可以与任何欧几里得空间一起使用，或者更技术地说，可以与具有内积的实数域上的任何有限维向量空间一起使用。笛卡尔张量的使用出现在物理和工程中，例如柯西应力张量和刚体动力学中的惯性矩张量。有时，广义曲线坐标很方便，如在高变形连续介质力学中，甚至是必要的，如在广义相对论中。
In geometry and linear algebra, a Cartesian tensor uses an orthonormal basis to represent a tensor in a Euclidean space in the form of components. Converting a tensor's components from one such basis to another is done through an orthogonal transformation. The most familiar coordinate systems are the two-dimensional and three-dimensional Cartesian coordinate systems. Cartesian tensors may be used with any Euclidean space, or more technically, any finite-dimensional vector space over the field of real numbers that has an inner product. Use of Cartesian tensors occurs in physics and engineering, such as with the Cauchy stress tensor and the moment of inertia tensor in rigid body dynamics. Sometimes general curvilinear coordinates are convenient, as in high-deformation continuum mechanics, or even necessary, as in general relativity.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，矩阵范畴通常表示为 Ma t {\displaystyle \mathbf {Mat} } ，其对象是自然数，其态射是矩阵，其组合由矩阵乘法给出。
In mathematics, the category of matrices, often denoted M a t {\displaystyle \mathbf {Mat} } , is the category whose objects are natural numbers and whose morphisms are matrices, with composition given by matrix multiplication.
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View content license ↗ 在代数中，给定一个环 R {\displaystyle R} ， R {\displaystyle R} 上的左模范畴是其对象都是 R {\displaystyle R} 上的左模且其态射都是左 R {\displaystyle R} 模之间的模同态的范畴。例如，当 R {\displaystyle R} 是整数环 Z {\displaystyle \mathbb {Z} } 时，它与阿贝尔群的范畴是一样的。右模块的类别以类似的方式定义。人们还可以定义环 R {\displaystyle R} 上的双模类别，但该类别相当于 R {\displaystyle R} 的包络代数（或与之相反）上的左（或右）模类别。
In algebra, given a ring R {\displaystyle R} , the category of left modules over R {\displaystyle R} is the category whose objects are all left modules over R {\displaystyle R} and whose morphisms are all module homomorphisms between left R {\displaystyle R} -modules. For example, when R {\displaystyle R} is the ring of integers Z {\displaystyle \mathbb {Z} } , it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring R {\displaystyle R} but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R {\displaystyle R} (or over the opposite of that).
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 柯西-施瓦茨不等式（也称为柯西-布尼亚科夫斯基-施瓦茨不等式）是内积空间中两个向量之间的内积绝对值的上限，以向量范数的乘积表示。它被认为是数学中最重要和最广泛使用的不等式之一。向量的内积可以描述有限和（通过有限维向量空间）、无限级数（通过序列空间中的向量）和积分（通过希尔伯特空间中的向量）。总和不等式由 Augustin-Louis Cauchy (1821) 发表。相应的积分不等式由 Viktor Bunyakovsky (1859) 和 Hermann Schwarz (1888) 发表。施瓦茨给出了积分版本的现代证明。
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics. Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by Augustin-Louis Cauchy (1821). The corresponding inequality for integrals was published by Viktor Bunyakovsky (1859) and Hermann Schwarz (1888). Schwarz gave the modern proof of the integral version.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，特别是在线性代数和矩阵论中，中心对称矩阵是关于其中心对称的矩阵。
In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，有限维 n 的向量空间的有序基允许通过坐标向量唯一地表示向量空间的任何元素，坐标向量是称为坐标的 n 标量的有限序列。如果考虑两个不同的基，则在一个基上表示向量 v 的坐标向量通常不同于在另一基上表示 v 的坐标向量。基础的改变包括将以相对于一个基础的坐标表示的每个断言转换为以相对于另一基础的坐标表示的断言。这种转换源自基差变化公式，该公式用相对于另一个基差的坐标来表达相对于一个基差的坐标。
In mathematics, an ordered basis of a vector space of finite dimension n allows representing uniquely any element of the vector space by a coordinate vector, which is a finite sequence of n scalars called coordinates. If two different bases are considered, the coordinate vector that represents a vector v on one basis is, in general, different from the coordinate vector that represents v on the other basis. A change of basis consists of converting every assertion expressed in terms of coordinates relative to one basis into an assertion expressed in terms of coordinates relative to the other basis. Such a conversion results from the change-of-basis formula, which expresses the coordinates relative to one basis in terms of the coordinates relative to the other basis.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在线性代数中，方阵的特征多项式是在矩阵相似性下不变的、以特征值为根的多项式。它的系数之间具有矩阵的行列式和迹。有限维向量空间的自同态的特征多项式是该自同态矩阵在任何基上的特征多项式（即特征多项式不依赖于基的选择）。特征方程又称行列式方程，是将特征多项式设为零得到的方程。在谱图论中，图的特征多项式就是其邻接矩阵的特征多项式。
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is, the characteristic polynomial does not depend on the choice of a basis). The characteristic equation, also known as the determinantal equation, is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory, the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，完全正映射的 Choi 定理是对有限维（矩阵）C* 代数之间的完全正映射进行分类的结果。这个 1975 年定理是由 Man-Duen Choi 提出的。 Choi 定理的无限维代数推广被称为完全正映射的 Belavkin 的“Radon-Nikodym”定理。
In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. This 1975 theorem is due to Man-Duen Choi. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's "Radon–Nikodym" theorem for completely positive maps.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 线性组合（英语：Linear combination）是线性代数中具有如下形式的表达式。其中 v i {\displaystyle v_{i}} 为任意类型的项， a i {\displaystyle a_{i}} 为标量。这些标量称为线性组合的系数或权。 w = a 1 v 1 + a 2 v 2 + a 3 v 3 + ⋯ + a n v n {\displaystyle w=a_{1}v_{1}+a_{2}v_{2}+a_{3}v_{3}+\cdots +a_{n}v_{n}}
In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
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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 ↗ 在线性代数中，向量空间或模的元素之间的线性关系或简单关系是将这些元素作为解的线性方程。
In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在量子力学的数学物理中，刘维尔空间也称为线空间，是希尔伯特空间上的算子空间。刘维尔空间本身就是希尔伯特-施密特内积下的希尔伯特空间。抽象地讲，刘维尔空间等价于（等距同构）希尔伯特空间及其对偶的张量积。在刘维尔空间中组织计算的常见计算技术是矢量化。刘维尔空间是密度算子形式主义的基础，是开放量子系统研究中的常用计算技术。
In the mathematical physics of quantum mechanics, Liouville space, also known as line space, is the space of operators on Hilbert space. Liouville space is itself a Hilbert space under the Hilbert-Schmidt inner product. Abstractly, Liouville space is equivalent (isometrically isomorphic) to the tensor product of a Hilbert space with its dual. A common computational technique to organize computations in Liouville space is vectorization. Liouville space underlies the density operator formalism and is a common computation technique in the study of open quantum systems.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，如果空间 V {\displaystyle V} 是一系列有限维 f {\displaystyle f} 不变子空间的并集，则线性算子 f : V → V {\displaystyle f:V\to V} 被称为局部有限。
In mathematics, a linear operator f : V → V {\displaystyle f:V\to V} is called locally finite if the space V {\displaystyle V} is the union of a family of finite-dimensional f {\displaystyle f} -invariant subspaces.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在数学中，Loewner 阶是由正半定矩阵的凸锥定义的偏阶。该顺序通常用于将单调和凹/凸标量函数的定义推广为单调和凹/凸埃尔米特值函数。这些函数自然出现在矩阵和算子理论中，并在物理和工程的许多领域都有应用。
In mathematics, Loewner order is the partial order defined by the convex cone of positive semi-definite matrices. This order is usually employed to generalize the definitions of monotone and concave/convex scalar functions to monotone and concave/convex Hermitian valued functions. These functions arise naturally in matrix and operator theory and have applications in many areas of physics and engineering.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ Maple 是一种符号和数值计算环境，也是一种多范式编程语言。它涵盖了技术计算的多个领域，例如符号数学、数值分析、数据处理、可视化等。工具箱 MapleSim 添加了多域物理建模和代码生成的功能。 Maple 的符号计算能力包括通用计算机代数系统的能力。例如，它可以操纵数学表达式并找到某些问题的符号解，例如由常微分方程和偏微分方程引起的问题。 Maple 由加拿大软件公司 Maplesoft 进行商业开发。 “Maple”这个名字是指该软件的加拿大传统。
Maple is a symbolic and numeric computing environment as well as a multi-paradigm programming language. It covers several areas of technical computing, such as symbolic mathematics, numerical analysis, data processing, visualization, and others. A toolbox, MapleSim, adds functionality for multidomain physical modeling and code generation. Maple's capacity for symbolic computing include those of a general-purpose computer algebra system. For instance, it can manipulate mathematical expressions and find symbolic solutions to certain problems, such as those arising from ordinary and partial differential equations. Maple is developed commercially by the Canadian software company Maplesoft. The name 'Maple' is a reference to the software's Canadian heritage.
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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. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
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