跟踪函数的期望值和方差

13

为随机变量，和一个半正定矩阵：是否有预期的数值的简化表达式，和方差，？请注意，不是随机变量。 $X \in \mathbb{R}^h$ $A$ $\mathop {\mathbb E}[Tr(X^TAX)]$ $Var[Tr(X^TAX)]$ $A$

variance mathematical-statistics expected-value random-matrix

— 灵车
source

Answers:

14

由于 $X^TAX$ 是标量，因此

Tr (X^{T} A X) = X^{T} A X = Tr (A X X^{T})

$\text{Tr}(X^TAX)= X^TAX = \text{Tr}(AXX^T)$ 因此

E (X^{T} A X) = E (Tr (A X X^{T})) = Tr (E (A X X^{T})) = Tr (A E (X X^{T}))

$\text{E}(X^TAX) = \text{E}(\text{Tr}(AXX^T)) = \text{Tr}(\text{E} (A XX^T)) = \text{Tr}(A\text{E}(XX^T))$

$X$

— 凯捷蒂尔·哈沃森
source

T r (X^{T} A X) = X^{T} A X

$Tr(X^T AX) = X^T AX$

1

X^{T} A X

$X^TAX$

By using our site, you acknowledge that you have read and understand our Cookie Policy and Privacy Policy.

Licensed under cc by-sa 3.0 with attribution required.