JordanDecomposition
✖
JordanDecomposition
范例
打开所有单元关闭所有单元基本范例 (1)常见实例总结
范围 (10)标准用法实例范围调查
基本用法 (6)

https://wolfram.com/xid/0b0mapsa4pe-zu0gyk


https://wolfram.com/xid/0b0mapsa4pe-h1ahoy


https://wolfram.com/xid/0b0mapsa4pe-zrmlpw

https://wolfram.com/xid/0b0mapsa4pe-7iibn6

具有亏特征空间 (deficient eigenspace) 的精确矩阵的约旦分解:

https://wolfram.com/xid/0b0mapsa4pe-vyrris

https://wolfram.com/xid/0b0mapsa4pe-f1rxq4

https://wolfram.com/xid/0b0mapsa4pe-i9tzbx


https://wolfram.com/xid/0b0mapsa4pe-n6ilek

因此 的第三列是一个广义的特征向量,其中
给出
,而不是
:

https://wolfram.com/xid/0b0mapsa4pe-n5y3nu


https://wolfram.com/xid/0b0mapsa4pe-fdp2bk


https://wolfram.com/xid/0b0mapsa4pe-kh8lz1


https://wolfram.com/xid/0b0mapsa4pe-madl2m

https://wolfram.com/xid/0b0mapsa4pe-0ipkhx

特殊矩阵 (4)

https://wolfram.com/xid/0b0mapsa4pe-g3c1vo


https://wolfram.com/xid/0b0mapsa4pe-6pl8u9


https://wolfram.com/xid/0b0mapsa4pe-wrpyns


https://wolfram.com/xid/0b0mapsa4pe-30cyz3


https://wolfram.com/xid/0b0mapsa4pe-k0xjhe


https://wolfram.com/xid/0b0mapsa4pe-pthoj7

IdentityMatrix 是约旦标准型:

https://wolfram.com/xid/0b0mapsa4pe-4x01lt

https://wolfram.com/xid/0b0mapsa4pe-oy5mpe


https://wolfram.com/xid/0b0mapsa4pe-frdsxs


https://wolfram.com/xid/0b0mapsa4pe-f63083

HilbertMatrix 的约旦分解:

https://wolfram.com/xid/0b0mapsa4pe-mqxkx3

应用 (12)用该函数可以解决的问题范例
广义特征向量和可对角化性 (4)
对于矩阵 ,根据真实特征向量和广义特征向量解释约旦分解的
矩阵的列:

https://wolfram.com/xid/0b0mapsa4pe-6gbrq4

https://wolfram.com/xid/0b0mapsa4pe-gfzptm

在对应的 — 第 1、3 和 4 列— 的对角线上没有
的
的列是真正的特征向量,其中
:

https://wolfram.com/xid/0b0mapsa4pe-i7a8ov


https://wolfram.com/xid/0b0mapsa4pe-6evvit

证明下面的矩阵只有一个特征向量,但它有一个完整的广义特征向量链,构成了 的基:

https://wolfram.com/xid/0b0mapsa4pe-odtwt9
Eigensystem 显示 84 是重数为 4 的特征值,它只有一个独立的特征向量:

https://wolfram.com/xid/0b0mapsa4pe-dnjjd3

JordanDecomposition 的 矩阵的第一列是找到的一个特征向量:

https://wolfram.com/xid/0b0mapsa4pe-bwrb23


https://wolfram.com/xid/0b0mapsa4pe-qc1zvl

由于 的 NullSpace 为空,其列构成
的基:

https://wolfram.com/xid/0b0mapsa4pe-hkktev

有且仅有当方块矩阵的 矩阵为对角矩阵时,其有完整的特征向量组,因此该矩阵可对角化:

https://wolfram.com/xid/0b0mapsa4pe-h6peol

https://wolfram.com/xid/0b0mapsa4pe-16pr0x

使用 DiagonalizableMatrixQ 进行验证:

https://wolfram.com/xid/0b0mapsa4pe-70uv9z


https://wolfram.com/xid/0b0mapsa4pe-oykxfq


https://wolfram.com/xid/0b0mapsa4pe-7q1re9


https://wolfram.com/xid/0b0mapsa4pe-vxl6ny

https://wolfram.com/xid/0b0mapsa4pe-s3q83e

可用 JordanDecomposition 将 写成可对角化矩阵和幂零矩阵的和:

https://wolfram.com/xid/0b0mapsa4pe-jpeinx


https://wolfram.com/xid/0b0mapsa4pe-wf0ucl


https://wolfram.com/xid/0b0mapsa4pe-kcsi17


https://wolfram.com/xid/0b0mapsa4pe-qgcrxh


https://wolfram.com/xid/0b0mapsa4pe-us5pk1


https://wolfram.com/xid/0b0mapsa4pe-0a8wrj


https://wolfram.com/xid/0b0mapsa4pe-fpmzz2

对角化 (4)
对于可对角化矩阵,约旦分解直接给出其对角化为 . 将其应用于将矩阵
对角化:

https://wolfram.com/xid/0b0mapsa4pe-7gdki7

https://wolfram.com/xid/0b0mapsa4pe-q713gz


https://wolfram.com/xid/0b0mapsa4pe-jt277v

令 为线性变换,其标准矩阵由矩阵
给出. 求得
的基
,在该基
中
表示的属性为对角矩阵:

https://wolfram.com/xid/0b0mapsa4pe-hld5cy

https://wolfram.com/xid/0b0mapsa4pe-o61dga

令 由特征向量组成,即
的列. 当
从
坐标转换为标准坐标时,其逆向转换为相反方向:

https://wolfram.com/xid/0b0mapsa4pe-6tp8ia


https://wolfram.com/xid/0b0mapsa4pe-gefesr


https://wolfram.com/xid/0b0mapsa4pe-ytgfvr

实值对称矩阵 可正交对角化为
,其中
为实值对角矩阵且
为正交. 验证以下矩阵是对称的,然后对其进行对角化:

https://wolfram.com/xid/0b0mapsa4pe-t0tayn


https://wolfram.com/xid/0b0mapsa4pe-d26vmx


https://wolfram.com/xid/0b0mapsa4pe-mbie39

https://wolfram.com/xid/0b0mapsa4pe-c0pq92


https://wolfram.com/xid/0b0mapsa4pe-2tw7d1


https://wolfram.com/xid/0b0mapsa4pe-cuel5e


https://wolfram.com/xid/0b0mapsa4pe-4fffl2

若 则矩阵被称为正规矩阵. 正规矩阵是最通用的一种矩阵,可以通过酉变换对角化. 所有实对称矩阵
都是正规矩阵,因为等式的两边都是
:

https://wolfram.com/xid/0b0mapsa4pe-6b09sy


https://wolfram.com/xid/0b0mapsa4pe-exqe2b


https://wolfram.com/xid/0b0mapsa4pe-wqtvhs

使用 NormalMatrixQ 验证:

https://wolfram.com/xid/0b0mapsa4pe-46r3j6


https://wolfram.com/xid/0b0mapsa4pe-0bbmuk

https://wolfram.com/xid/0b0mapsa4pe-8d13py


https://wolfram.com/xid/0b0mapsa4pe-3xuanx


https://wolfram.com/xid/0b0mapsa4pe-64w99o


https://wolfram.com/xid/0b0mapsa4pe-2hpiyq

矩阵函数和动态系统 (4)

https://wolfram.com/xid/0b0mapsa4pe-jqh2nk

https://wolfram.com/xid/0b0mapsa4pe-lo6s0e


https://wolfram.com/xid/0b0mapsa4pe-x8d19i

然后 . 由于
是上三角形且接近对角线,因此对角线项的乘方为
,项
变为
:

https://wolfram.com/xid/0b0mapsa4pe-2mtd8o


https://wolfram.com/xid/0b0mapsa4pe-1ms4pa

通过 MatrixPower 的直接计算进行验证:

https://wolfram.com/xid/0b0mapsa4pe-ut4jb3

应用指数的幂级数,对角线项变成 ,而非对角线项只是重新索引的指数和. 因此,它也变成了
:

https://wolfram.com/xid/0b0mapsa4pe-x4ujj6


https://wolfram.com/xid/0b0mapsa4pe-v3k1is

通过 MatrixExp 的计算进行验证:

https://wolfram.com/xid/0b0mapsa4pe-3chst3

验证约旦矩阵的方程 ,该约旦矩阵由下列矩阵
和函数
,
和
的单链组成:

https://wolfram.com/xid/0b0mapsa4pe-qbmjxa


https://wolfram.com/xid/0b0mapsa4pe-p6jlws

使用 MatrixFunction 验证该计算:

https://wolfram.com/xid/0b0mapsa4pe-405v2o


https://wolfram.com/xid/0b0mapsa4pe-pb24s4

MatrixFunction 验证了该结果:

https://wolfram.com/xid/0b0mapsa4pe-v2l3rx

由于 有参数,需要使用 D 而非 Derivative,并在
中代换:

https://wolfram.com/xid/0b0mapsa4pe-x8nm0q

同样,当使用 Function 输入 时,MatrixFunction 确认结果:

https://wolfram.com/xid/0b0mapsa4pe-j6f7ys


https://wolfram.com/xid/0b0mapsa4pe-uoeghr


https://wolfram.com/xid/0b0mapsa4pe-7s4qu6


https://wolfram.com/xid/0b0mapsa4pe-p91gkv


https://wolfram.com/xid/0b0mapsa4pe-flftlr

使用 DSolveValue 验证解:

https://wolfram.com/xid/0b0mapsa4pe-neb1o5


https://wolfram.com/xid/0b0mapsa4pe-pp2m0k

https://wolfram.com/xid/0b0mapsa4pe-hcmla7


https://wolfram.com/xid/0b0mapsa4pe-q30bvq


https://wolfram.com/xid/0b0mapsa4pe-ow6p6z


https://wolfram.com/xid/0b0mapsa4pe-7yd97k

属性和关系 (10)函数的属性及与其他函数的关联
JordanDecomposition[m] 给出 m 的矩阵分解为 s.j.Inverse[s]:

https://wolfram.com/xid/0b0mapsa4pe-nkzr7p

https://wolfram.com/xid/0b0mapsa4pe-qkayu8


https://wolfram.com/xid/0b0mapsa4pe-eu2pjn

m 等于 s.j.Inverse[s]:

https://wolfram.com/xid/0b0mapsa4pe-jm2k7d


https://wolfram.com/xid/0b0mapsa4pe-fulz9x


https://wolfram.com/xid/0b0mapsa4pe-4mwr0y

https://wolfram.com/xid/0b0mapsa4pe-5dg1a0


https://wolfram.com/xid/0b0mapsa4pe-1vzwst

https://wolfram.com/xid/0b0mapsa4pe-wt6fit

如果 m 是对角化的,Jordan 分解实际上和 Eigensystem 相同:

https://wolfram.com/xid/0b0mapsa4pe-gk9m34

https://wolfram.com/xid/0b0mapsa4pe-3asj2m


https://wolfram.com/xid/0b0mapsa4pe-ll6wi1


https://wolfram.com/xid/0b0mapsa4pe-onu3q4


https://wolfram.com/xid/0b0mapsa4pe-k3rk6n


https://wolfram.com/xid/0b0mapsa4pe-flnp05

对于可对角化矩阵, JordanDecomposition 将函数应用简化为特征值的应用:

https://wolfram.com/xid/0b0mapsa4pe-6ufrqm

https://wolfram.com/xid/0b0mapsa4pe-tlnt0f

使用 MatrixExp 计算矩阵指数:

https://wolfram.com/xid/0b0mapsa4pe-c6fiyq


https://wolfram.com/xid/0b0mapsa4pe-y2pltu

对于不可对角化矩阵,约旦分解将函数应用限制在每个广义特征向量链上:

https://wolfram.com/xid/0b0mapsa4pe-kt5xge

https://wolfram.com/xid/0b0mapsa4pe-wcrgtj

https://wolfram.com/xid/0b0mapsa4pe-qwbsg3

对于 j 在对角线上方有 1 的列,函数应用仅扩展到对角线上方:

https://wolfram.com/xid/0b0mapsa4pe-e2owjo


https://wolfram.com/xid/0b0mapsa4pe-hfvq78


https://wolfram.com/xid/0b0mapsa4pe-40f8st


https://wolfram.com/xid/0b0mapsa4pe-zw892l


https://wolfram.com/xid/0b0mapsa4pe-1n3v4a


https://wolfram.com/xid/0b0mapsa4pe-2d2hq1


https://wolfram.com/xid/0b0mapsa4pe-xvt54m


https://wolfram.com/xid/0b0mapsa4pe-nm40tw


https://wolfram.com/xid/0b0mapsa4pe-wwepwu


https://wolfram.com/xid/0b0mapsa4pe-4w1n61


https://wolfram.com/xid/0b0mapsa4pe-3y0ct4


https://wolfram.com/xid/0b0mapsa4pe-mclbfd


https://wolfram.com/xid/0b0mapsa4pe-wz0l89


https://wolfram.com/xid/0b0mapsa4pe-jxr4wq


https://wolfram.com/xid/0b0mapsa4pe-45k79q


https://wolfram.com/xid/0b0mapsa4pe-mylxd7


https://wolfram.com/xid/0b0mapsa4pe-c1kzdd

正规矩阵 的 SchurDecomposition[n,RealBlockDiagonalFormFalse]:

https://wolfram.com/xid/0b0mapsa4pe-df3aev


https://wolfram.com/xid/0b0mapsa4pe-r3cio0


https://wolfram.com/xid/0b0mapsa4pe-j7q4dl


https://wolfram.com/xid/0b0mapsa4pe-3ehy88

为验证 q 是否具有特征向量作为列,将每列的第一个条目设置为 1. 可以消除 q 和 s 之间的相位差:

https://wolfram.com/xid/0b0mapsa4pe-jkbsky

可能存在的问题 (2)常见隐患和异常行为

https://wolfram.com/xid/0b0mapsa4pe-ph4cew

https://wolfram.com/xid/0b0mapsa4pe-lct9tf

https://wolfram.com/xid/0b0mapsa4pe-qcoehp


https://wolfram.com/xid/0b0mapsa4pe-ecnkgj


https://wolfram.com/xid/0b0mapsa4pe-3csgas


https://wolfram.com/xid/0b0mapsa4pe-bxlmii


https://wolfram.com/xid/0b0mapsa4pe-jsnhn2
由于数值四舍五入,2. 处的缺陷特征空间被分成两个独立的特征空间:

https://wolfram.com/xid/0b0mapsa4pe-41jhie


https://wolfram.com/xid/0b0mapsa4pe-prbuv6

Wolfram Research (1996),JordanDecomposition,Wolfram 语言函数,https://reference.wolfram.com/language/ref/JordanDecomposition.html (更新于 2010 年).
文本
Wolfram Research (1996),JordanDecomposition,Wolfram 语言函数,https://reference.wolfram.com/language/ref/JordanDecomposition.html (更新于 2010 年).
Wolfram Research (1996),JordanDecomposition,Wolfram 语言函数,https://reference.wolfram.com/language/ref/JordanDecomposition.html (更新于 2010 年).
CMS
Wolfram 语言. 1996. "JordanDecomposition." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2010. https://reference.wolfram.com/language/ref/JordanDecomposition.html.
Wolfram 语言. 1996. "JordanDecomposition." Wolfram 语言与系统参考资料中心. Wolfram Research. 最新版本 2010. https://reference.wolfram.com/language/ref/JordanDecomposition.html.
APA
Wolfram 语言. (1996). JordanDecomposition. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/JordanDecomposition.html 年
Wolfram 语言. (1996). JordanDecomposition. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/JordanDecomposition.html 年
BibTeX
@misc{reference.wolfram_2025_jordandecomposition, author="Wolfram Research", title="{JordanDecomposition}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/JordanDecomposition.html}", note=[Accessed: 02-April-2025
]}
BibLaTeX
@online{reference.wolfram_2025_jordandecomposition, organization={Wolfram Research}, title={JordanDecomposition}, year={2010}, url={https://reference.wolfram.com/language/ref/JordanDecomposition.html}, note=[Accessed: 02-April-2025
]}