定義 —
設
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Ω
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Σ
,
P
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{\displaystyle (\Omega ,\,\Sigma ,\,P)}
是機率空間，
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{\displaystyle X=\{x_{i}\}_{i=1}^{m}}
与
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{\displaystyle Y=\{y_{i}\}_{j=1}^{n}}
是定義在
Ω
{\displaystyle \Omega }
上的兩列实数随机变量序列
若二者对应的期望值分别为：
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Ω
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d
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{\displaystyle E(x_{i})=\int _{\Omega }x_{i}\,dP=\mu _{i}}
E
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Ω
y
j
d
P
=
ν
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{\displaystyle E(y_{j})=\int _{\Omega }y_{j}\,dP=\nu _{j}}
則这两列隨機变量间的协方差矩阵为：
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:=
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{\displaystyle \operatorname {\mathbf {cov} } (X,Y):={\left[\,\operatorname {cov} (x_{i},y_{j})\,\right]}_{m\times n}={{\bigg [}\,\operatorname {E} [(x_{i}-\mu _{i})(y_{j}-\nu _{j})]\,{\bigg ]}}_{m\times n}}
將之以矩形表示的話就是：
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{\displaystyle \operatorname {\mathbf {cov} } (X,Y)={\begin{bmatrix}\operatorname {cov} (x_{1},y_{1})&\operatorname {cov} (x_{1},y_{2})&\cdots &\operatorname {cov} (x_{1},y_{n})\\\operatorname {cov} (x_{2},y_{1})&\operatorname {cov} (x_{2},y_{2})&\cdots &\operatorname {cov} (x_{2},y_{n})\\\vdots &\vdots &\ddots &\vdots \\\operatorname {cov} (x_{m},y_{1})&\operatorname {cov} (x_{m},y_{2})&\cdots &\operatorname {cov} (x_{m},y_{n})\end{bmatrix}}}
=
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{\displaystyle ={\begin{bmatrix}\mathrm {E} [(x_{1}-\mu _{1})(y_{1}-\nu _{1})]&\mathrm {E} [(x_{1}-\mu _{1})(y_{2}-\nu _{2})]&\cdots &\mathrm {E} [(x_{1}-\mu _{1})(y_{n}-\nu _{n})]\\\mathrm {E} [(x_{2}-\mu _{2})(y_{1}-\nu _{1})]&\mathrm {E} [(x_{2}-\mu _{2})(y_{2}-\nu _{2})]&\cdots &\mathrm {E} [(x_{2}-\mu _{2})(y_{n}-\nu _{n})]\\\vdots &\vdots &\ddots &\vdots \\\mathrm {E} [(x_{m}-\mu _{m})(y_{1}-\nu _{1})]&\mathrm {E} [(x_{m}-\mu _{m})(y_{2}-\nu _{2})]&\cdots &\mathrm {E} [(x_{m}-\mu _{m})(y_{n}-\nu _{n})]\end{bmatrix}}}
根據測度積分的線性性質，协方差矩阵還可以進一步化簡為：
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{\displaystyle \operatorname {\mathbf {cov} } (X,Y)={\left[\,\operatorname {E} (x_{i}y_{j})-\mu _{i}\nu _{j}\,\right]}_{n\times n}}
矩陣表示法
编辑
以上定義所述的隨機變數序列
X
{\displaystyle X}
和
Y
{\displaystyle Y}
，也可分別以用行向量
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[
x
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]
m
{\displaystyle \mathbf {X} :={\left[x_{i}\right]}_{m}}
與
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n
{\displaystyle \mathbf {Y} :={\left[y_{j}\right]}_{n}}
表示，換句話說：
X
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[
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1
x
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⋮
x
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]
{\displaystyle \mathbf {X} :={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{m}\end{bmatrix}}}
Y
:=
[
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y
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⋮
y
n
]
{\displaystyle \mathbf {Y} :={\begin{bmatrix}y_{1}\\y_{2}\\\vdots \\y_{n}\end{bmatrix}}}
這樣的話，對於
m
×
n
{\displaystyle m\times n}
個定義在
Ω
{\displaystyle \Omega }
上的隨機變數
a
i
j
{\displaystyle a_{ij}}
所組成的矩陣
A
=
[
a
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j
]
m
×
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{\displaystyle \mathbf {A} ={\left[\,a_{ij}\,\right]}_{m\times n}}
， 定義：
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{\displaystyle \mathrm {E} [\mathbf {A} ]:={\left[\,\operatorname {E} (a_{ij})\,\right]}_{m\times n}}
也就是說
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:=
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{\displaystyle \mathrm {E} [\mathbf {A} ]:={\begin{bmatrix}\operatorname {E} (a_{11})&\operatorname {E} (a_{12})&\cdots &\operatorname {E} (a_{1n})\\\operatorname {E} (a_{21})&\operatorname {E} (a_{22})&\cdots &\operatorname {E} (a_{2n})\\\vdots &\vdots &\ddots &\vdots \\\operatorname {E} (a_{m1})&\operatorname {E} (a_{m2})&\cdots &\operatorname {E} (a_{mn})\end{bmatrix}}}
那上小節定義的协方差矩阵就可以記为：
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{\displaystyle \operatorname {\mathbf {cov} } (X,Y)=\mathrm {E} \left[\left(\mathbf {X} -\mathrm {E} [\mathbf {X} ]\right)\left(\mathbf {Y} -\mathrm {E} [\mathbf {Y} ]\right)^{\rm {T}}\right]}
所以协方差矩阵也可對
X
{\displaystyle \mathbf {X} }
與
Y
{\displaystyle \mathbf {Y} }
來定義：
c
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:=
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)
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{\displaystyle \operatorname {\mathbf {cov} } (\mathbf {X} ,\mathbf {Y} ):=\mathrm {E} \left[\left(\mathbf {X} -\mathrm {E} [\mathbf {X} ]\right)\left(\mathbf {Y} -\mathrm {E} [\mathbf {Y} ]\right)^{\rm {T}}\right]}
术语与符号分歧
编辑
也有人把以下的
Σ
X
{\displaystyle \mathbf {\Sigma } _{X}}
稱為协方差矩阵：
Σ
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:=
[
cov
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,
x
j
)
]
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×
m
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{\displaystyle {\begin{aligned}\mathbf {\Sigma } _{X}&:={\left[\operatorname {cov} (x_{i},x_{j})\right]}_{m\times m}\\&=\operatorname {\mathbf {cov} } (X,X)\end{aligned}}}
但本頁面沿用威廉·费勒的说法，把
Σ
X
{\displaystyle \mathbf {\Sigma } _{X}}
稱為
X
{\displaystyle X}
的方差（variance of random vector），來跟
c
o
v
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,
Y
)
{\displaystyle \operatorname {\mathbf {cov} } (X,Y)}
作區別。這是因為：
cov
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=
E
[
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x
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μ
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=
var
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)
{\displaystyle \operatorname {cov} (x_{i},x_{i})=\operatorname {E} [{(x_{i}-\mu _{i})}^{2}]=\operatorname {var} (x_{i})}
換句話說，
Σ
X
{\displaystyle \mathbf {\Sigma } _{X}}
的對角線由隨機變數
x
i
{\displaystyle x_{i}}
的方差所組成。據此，也有人也把
c
o
v
(
X
,
Y
)
{\displaystyle \operatorname {\mathbf {cov} } (X,Y)}
稱為方差-协方差矩阵（variance–covariance matrix）。
更有人因為方差和离差的相關性，含混的將
c
o
v
(
X
,
Y
)
{\displaystyle \operatorname {\mathbf {cov} } (X,Y)}
稱為离差矩阵。