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CircularOrthogonalMatrixDistribution[n]

表示一个圆正交矩阵分布，其矩阵维度为 {n,n}.

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Details and Options Details and Options
背景
范例  
基本范例  
范围  
应用  
属性和关系  
可能存在的问题  
参见
相关指南
历史
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CircularOrthogonalMatrixDistribution

CircularOrthogonalMatrixDistribution[n]

表示一个圆正交矩阵分布，其矩阵维度为 {n,n}.

更多信息

背景

范例

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基本范例  (2)

生成一个伪随机 COE 矩阵：

Wolfram Language code: m = RandomVariate[CircularOrthogonalMatrixDistribution[3]]

它是一个酉矩阵，也是一个对称矩阵：

Wolfram Language code: UnitaryMatrixQ[m]∧SymmetricMatrixQ[m]

用 MatrixPropertyDistribution 在球面上取一个随机点：

Wolfram Language code: mat2vec[m_] := Re[Table[Tr[m.PauliMatrix[k]], {k, 3}]] / 2
Wolfram Language code: dist = MatrixPropertyDistribution[mat2vec[u.PauliMatrix[3].u^], uCircularOrthogonalMatrixDistribution[2]];
Wolfram Language code: RandomVariate[dist]∈Sphere[]

可见分布聚集在 轴附近：

Wolfram Language code: Point[RandomVariate[dist, 1500]]//Graphics3D

范围  (3)

生成单个伪随机矩阵：

Wolfram Language code: RandomVariate[CircularOrthogonalMatrixDistribution[2]]

生成一组伪随机矩阵：

Wolfram Language code: RandomVariate[CircularOrthogonalMatrixDistribution[2], 2]

用数值法计算统计特性：

Wolfram Language code: gap𝒟 = MatrixPropertyDistribution[Max[Differences[Sort[Arg[Eigenvalues[𝓂]]]]], 𝓂CircularOrthogonalMatrixDistribution[10]];
Wolfram Language code: NExpectation[δ Sin[δ], δgap𝒟]

应用  (2)

定义随机矩阵特征值的复参数的分布：

Wolfram Language code: evs𝒟 = MatrixPropertyDistribution[Arg[Eigenvalues[x]], xCircularOrthogonalMatrixDistribution[2]];

对随机排列的特征值的相位进行抽样：

Wolfram Language code: φs = RandomSample /@ RandomVariate[evs𝒟, 10 ^ 5];

可视化联合相位分布和 PDF 解析式：

Wolfram Language code: Show[ Histogram3D[φs, {-Pi, Pi, 0.2}, PDF], Plot3D[(1/16Pi)Abs[Exp[I ϕ1] - Exp[I ϕ2]], {ϕ1, -Pi, Pi}, {ϕ2, -Pi, Pi}, PlotStyle -> None, MeshStyle -> Thick], ViewPoint -> {10, 10, 3}]

CircularOrthogonalMatrixDistribution 的特征值的联合分布也是逆温度 时 Dyson 库仑气体在圆上的 Boltzmann 分布. 系统中每个粒子的平均哈密顿量为（不包含动能项）：

Wolfram Language code: hamiltonian[Z_] := Module[{n = Length[Z]}, Log[n] / 2 - Sum[Total[Log[Abs[Differences[Z, 1, k]]]], {k, n - 1}] / n ]

定义随机 COE 矩阵上哈密顿量值的分布：

Wolfram Language code: h𝒟 = MatrixPropertyDistribution[hamiltonian[Eigenvalues[x]], xCircularOrthogonalMatrixDistribution[n]];

计算大小不同的系统的哈密顿量的抽样平均：

Wolfram Language code: means = Table[{n, Mean[RandomVariate[h𝒟, 1000]]}, {n, 10, 60, 10}]

绘制抽样平均并将其与热力学极值进行比较：

Wolfram Language code: Show[ListPlot[means, PlotStyle -> {Orange, PointSize[Medium]}, Filling -> Axis, PlotRange -> {0, 0.37}, AxesLabel -> {"N", "〈H〉"}], Plot[(1/2) (Log[2] + PolyGamma[0, (3/2)]), {x, 0, 60}]]

属性和关系  (2)

特征值的相角的分布：

Wolfram Language code: dim = 300; args = RandomVariate[MatrixPropertyDistribution[Arg[Eigenvalues[x]], xCircularOrthogonalMatrixDistribution[dim]], 100];
Wolfram Language code: Histogram[Join@@args, {-Pi, Pi, 2Pi / 40}, PDF]

计算特征值之间的间距：

Wolfram Language code: diffs = Join@@Map[Composition[Differences, Sort], (args/2Pi)dim];

比较样本水平间距的直方图与解析形式，解析形式亦称为 Dyson 指标 的 Wigner 估测：

Wolfram Language code: WignerSurmisePDF[x_, β : 1] := (Pi / 2) x Exp[(-Pi / 4) x ^ 2]
Wolfram Language code: Show[Histogram[diffs, {0.1}, PDF], Plot[WignerSurmisePDF[x, 1], {x, 0, 3}]]

对于维度大小为 的 CircularOrthogonalMatrixDistribution 的特征向量，其元素的缩放模数服从 分布：

Wolfram Language code: 𝒟 = MatrixPropertyDistribution[Flatten[Abs[Eigenvectors[𝓂]] ^ 2], 𝓂CircularOrthogonalMatrixDistribution[50]];
Wolfram Language code: data = Normalize[Join@@RandomVariate[𝒟, 100], Mean];

将直方图与 ChiSquareDistribution 的 PDF 进行比较：

Wolfram Language code: Show[Histogram[data, "Scott", PDF], Plot[PDF[ChiSquareDistribution[1], x], {x, 0, 5}, PlotRange -> {{0, 5}, {0, 3}}]]

可能存在的问题  (2)

从 CircularOrthogonalMatrixDistribution 得来的矩阵不必是正交矩阵：

Wolfram Language code: RandomVariate[CircularOrthogonalMatrixDistribution[3]]
Wolfram Language code: OrthogonalMatrixQ[%]

用 CircularRealMatrixDistribution 来抽样一个随机正交实矩阵：

Wolfram Language code: RandomVariate[CircularRealMatrixDistribution[3]]
Wolfram Language code: OrthogonalMatrixQ[%]

可将从 CircularOrthogonalMatrixDistribution 得来的矩阵表示为矩阵 TemplateBox[{u}, Transpose].u，其中矩阵 服从 CircularUnitaryMatrixDistribution：

Wolfram Language code: u = RandomVariate[CircularUnitaryMatrixDistribution[2]];
Wolfram Language code: o = Transpose[u].u;
Wolfram Language code: UnitaryMatrixQ[o]∧SymmetricMatrixQ[o]
Wolfram Research (2015)，CircularOrthogonalMatrixDistribution，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CircularOrthogonalMatrixDistribution.html.

文本

Wolfram Research (2015)，CircularOrthogonalMatrixDistribution，Wolfram 语言函数，https://reference.wolfram.com/language/ref/CircularOrthogonalMatrixDistribution.html.

CMS

Wolfram 语言. 2015. "CircularOrthogonalMatrixDistribution." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/CircularOrthogonalMatrixDistribution.html.

APA

Wolfram 语言. (2015). CircularOrthogonalMatrixDistribution. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/CircularOrthogonalMatrixDistribution.html 年

BibTeX

@misc{reference.wolfram_2026_circularorthogonalmatrixdistribution, author="Wolfram Research", title="{CircularOrthogonalMatrixDistribution}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/CircularOrthogonalMatrixDistribution.html}", note=[Accessed: 15-September-2026]}

BibLaTeX

@online{reference.wolfram_2026_circularorthogonalmatrixdistribution, organization={Wolfram Research}, title={CircularOrthogonalMatrixDistribution}, year={2015}, url={https://reference.wolfram.com/language/ref/CircularOrthogonalMatrixDistribution.html}, note=[Accessed: 15-September-2026]}