Označimo z
X
→
{\displaystyle {\vec {X}}\,}
stolpični vektor
X
→
=
[
X
1
⋮
X
n
]
{\displaystyle {\vec {X}}={\begin{bmatrix}X_{1}\\\vdots \\X_{n}\end{bmatrix}}}
kjer so
X
n
{\displaystyle X_{n}\,}
posamezne komponente slučajne spremenljivke , ki imajo končno varianco .
Kovariančna matrika
Σ
{\displaystyle \Sigma \,}
, ki ima za elemente kovariance tako, da je
Σ
i
j
=
c
o
v
(
X
i
,
X
j
)
=
E
[
(
X
i
−
μ
i
)
(
X
j
−
μ
j
)
]
{\displaystyle \Sigma _{ij}=\mathrm {cov} (X_{i},X_{j})=\mathrm {E} {\begin{bmatrix}(X_{i}-\mu _{i})(X_{j}-\mu _{j})\end{bmatrix}}}
kjer je
μ
i
=
E
(
X
i
)
{\displaystyle \mu _{i}=\mathrm {E} (X_{i})\,}
pričakovana vrednost za i-to komponento vektorja
X
{\displaystyle X\,}
.
c
o
v
(
X
i
,
X
j
)
{\displaystyle \mathrm {cov} (X_{i},X_{j})\,}
kovarianca elementov
X
i
{\displaystyle X_{i}\,}
in
X
j
{\displaystyle X_{j}\,}
.Iz tega sledi, da kovariančno matriko lahko zapišemo kot
Σ
=
[
E
[
(
X
1
−
μ
1
)
(
X
1
−
μ
1
)
]
E
[
(
X
1
−
μ
1
)
(
X
2
−
μ
2
)
]
⋯
E
[
(
X
1
−
μ
1
)
(
X
n
−
μ
n
)
]
E
[
(
X
2
−
μ
2
)
(
X
1
−
μ
1
)
]
E
[
(
X
2
−
μ
2
)
(
X
2
−
μ
2
)
]
⋯
E
[
(
X
2
−
μ
2
)
(
X
n
−
μ
n
)
]
⋮
⋮
⋱
⋮
E
[
(
X
n
−
μ
n
)
(
X
1
−
μ
1
)
]
E
[
(
X
n
−
μ
n
)
(
X
2
−
μ
2
)
]
⋯
E
[
(
X
n
−
μ
n
)
(
X
n
−
μ
n
)
]
]
.
{\displaystyle \Sigma ={\begin{bmatrix}\mathrm {E} [(X_{1}-\mu _{1})(X_{1}-\mu _{1})]&\mathrm {E} [(X_{1}-\mu _{1})(X_{2}-\mu _{2})]&\cdots &\mathrm {E} [(X_{1}-\mu _{1})(X_{n}-\mu _{n})]\\\\\mathrm {E} [(X_{2}-\mu _{2})(X_{1}-\mu _{1})]&\mathrm {E} [(X_{2}-\mu _{2})(X_{2}-\mu _{2})]&\cdots &\mathrm {E} [(X_{2}-\mu _{2})(X_{n}-\mu _{n})]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(X_{n}-\mu _{n})(X_{1}-\mu _{1})]&\mathrm {E} [(X_{n}-\mu _{n})(X_{2}-\mu _{2})]&\cdots &\mathrm {E} [(X_{n}-\mu _{n})(X_{n}-\mu _{n})]\end{bmatrix}}.}
.Obratno matriko kovariančne matrike
Σ
−
1
{\displaystyle \Sigma ^{-1}\,}
imenujejo tudi matrika natančnosti .
Kovariančno matriko imenujemo tudi variančno-kovariančna matrika, ker velja
Σ
X
=
var
(
X
→
)
=
var
(
X
1
⋮
X
p
)
=
(
var
(
X
1
)
cov
(
X
1
X
2
)
⋯
cov
(
X
1
X
p
)
cov
(
X
2
X
1
)
⋱
⋯
⋮
⋮
⋮
⋱
⋮
cov
(
X
P
X
1
)
⋯
⋯
var
(
X
p
)
)
=
(
σ
x
1
2
σ
x
1
x
2
⋯
σ
x
1
x
p
σ
x
2
x
1
⋱
⋯
⋮
⋮
⋮
⋱
⋮
σ
x
p
x
1
⋯
⋯
σ
x
p
2
)
{\displaystyle \Sigma _{X}=\operatorname {var} ({\vec {X}})=\operatorname {var} {\begin{pmatrix}X_{1}\\\vdots \\X_{p}\end{pmatrix}}={\begin{pmatrix}\operatorname {var} (X_{1})&\operatorname {cov} (X_{1}X_{2})&\cdots &\operatorname {cov} (X_{1}X_{p})\\\operatorname {cov} (X_{2}X_{1})&\ddots &\cdots &\vdots \\\vdots &\vdots &\ddots &\vdots \\\operatorname {cov} (X_{P}X_{1})&\cdots &\cdots &\operatorname {var} (X_{p})\end{pmatrix}}={\begin{pmatrix}\sigma _{x_{1}}^{2}&\sigma _{x_{1}x_{2}}&\cdots &\sigma _{x_{1}x_{p}}\\\sigma _{x_{2}x_{1}}&\ddots &\cdots &\vdots \\\vdots &\vdots &\ddots &\vdots \\\sigma _{x_{p}x_{1}}&\cdots &\cdots &\sigma _{x_{p}}^{2}\end{pmatrix}}}
kjer je
var
(
X
→
)
{\displaystyle \operatorname {var} ({\vec {X}})\,}
varianca vektorja
X
→
{\displaystyle {\vec {X}}\,}
cov
{\displaystyle \operatorname {cov} \,}
kovarianca komponent
X
i
{\displaystyle X_{i}\,}
in
X
j
{\displaystyle X_{j}\,}
σ
n
{\displaystyle \sigma _{n}\,}
varianca n-te komponente vektorja (na glavni diagonali so same variance, izven diagonale pa so kovariance). Zaradi tega ima matrika tudi ime variančno-kovariančna matrika .