【6】随机阵的正态分布
对于一个矩阵
[X=
left(
egin{array}
{cccc}
x_{11} & x_{12} & dots & x_{1p}\
x_{21} & x_{22} & dots & x_{2p}\
vdots & vdots & & vdots \
x_{n1} & x_{n2} & dots & x_{np}\
end{array}
ight)=
left(
egin{array}
{c}
X'_{(1)}\
X'_{(2)}\
vdots\
X'_{(n)}
end{array}
ight)=(mathcal{X}_1,mathcal{X}_2dots,mathcal{X}_p)
]
设(X_{(i)}=(x_{i1},dots,x_{ip})'),（(i=1,dots,n)）为来自(p)元正态总体 (N_p(mu,Sigma)) 的独立同分布随机样本，记随机阵(X=(x_{ij})_{n imes p})，利用拉直运算,((mathbb{I}::=p维单位向量))及克罗内克积( Kronecker )运算，可知：
[Vec(X')sim N_{np}(mathbb{I}_notimesmu,I_notimesSigma)
]
事实上，
[Vec(X')= left( egin{array}{c} X_{(1)}\ vdots\ X_{(n)} end{array} ight)=(x_{11},dots,x_{1p},dots,x_{n1},dots,x_{np})' ]为一个(np)维的长向量，其联合密度函数为：
[egin{align} f(x_{(1)},dots,x_{(n)}) =&prod_{i=1}^nfrac1{(2pi)^{p/2}|Sigma|^{1/2}}exp{-frac12(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)}\ =&frac1{(2pi)^{np/2}|Sigma|^{n/2}}exp{-frac12sum_{i=1}^n(x_{(i)}-mu)'Sigma^{-1}(x_{(i)}-mu)}\ =&frac1{(2pi)^{np/2}|Sigma|^{n/2}}expleft{-frac12left( egin{array}{c} x_{(1)}-mu\ vdots\ x_{(n)}-mu end{array} ight)' left( egin{array}{ccc} Sigma&cdots&O\ vdots&&vdots\ O&cdots&Sigma\ end{array} ight)^{-1} left( egin{array}{c} x_{(1)}-mu\ vdots\ x_{(n)}-mu end{array} ight) ight}\ =&frac1{(2pi)^{np/2}|Sigma|^{n/2}}expleft{-frac12left( egin{array}{c} X-mathbb{I}_notimesmu end{array} ight)' (I_notimesSigma)^{-1} (X-mathbb{I}_notimesmu) ight}\ end{align} ]于是，当随机阵(X)按行进行拉直以后，若满足(Vec(X')sim N_{np}(mathbb{I}_notimesmu,I_notimesSigma))，则称其服从矩阵正态分布，记作：(Xsim N_{n imes p}(M，I_notimesSigma))
其中
[M=left( egin{array} {ccc} mu_1 & dots & mu_p\ vdots & & vdots \ mu_1 & dots & mu_p\ end{array} ight) =mathbb{I}_nmu'::= left( egin{array} {c} 1\vdots\1 end{array} ight)_{p imes1} (mu_1,dots,mu_p) ]则有
[Vec(M')=mathbb{I}_nmu=(mu_1,dots,mu_p,dots,mu_1,dots,mu_p)' ]于是
[Vec(X')sim N_{np}(Vec(M'),I_notimesSigma)quadleftrightarrowsquad Xsim N_{n imes p}(M，I_notimesSigma) ]
线性组合的性质
- 设(Xsim N_{n imes p}(M，I_notimesSigma)),令(Z=A_{k imes n}XB_{q imes p}'+D_{k imes q})，则：
[Zsim N_{k imes q}(AMB'+D，(AA')otimes(BSigma B'))
]