Cross-covariance matrix

Dependencies:

1. Random variable
2. Covariance of 2 random variables
3. Linearity of expectation for matrices

Let $X = [X_1, X_2, \ldots, X_m]$ and $Y = [Y_1, Y_2, \ldots, Y_n]$ be sequences of random variables. Then the cross-covariance matrix $\operatorname{Cov}(X, Y)$, an $m$-by-$n$ matrix, is defined as $\newcommand{\E}{\operatorname{E}}$ $\newcommand{\Cov}{\operatorname{Cov}}$ \[ \Cov(X, Y)_{i, j} = \Cov(X_i, Y_j) \] An alternative definition is \[ \Cov(X, Y) = \E((X-\E(X))(Y-\E(Y))^T) \] Another definition is \[ \Cov(X, Y) = \E(XY^T) - \E(X)\E(Y)^T \] These definitions are equivalent

Proof

\begin{align} \E((X-\E(X))(Y-\E(Y))^T)[i, j] &= \E((X-\E(X))_i (Y-\E(Y))_j) \\ &= \E((X_i-\E(X_i)) (Y_j-\E(Y_j))) = \Cov(X_i, Y_j) \end{align}

Using linearity of expectation for matrices, we get \begin{align} \E((X-\E(X))(Y-\E(Y))^T) &= \E(XY^T - \E(X)Y^T - X\E(Y)^T + \E(X)\E(Y)^T \\ &= \E(XY^T) - \E(X)\E(Y)^T \end{align}

Dependency for:

1. Covariance matrix

Info:

• Depth: 9
• Number of transitive dependencies: 22

Transitive dependencies:

1. /analysis/topological-space
2. /sets-and-relations/countable-set
3. /sets-and-relations/de-morgan-laws
4. /measure-theory/linearity-of-lebesgue-integral
5. /measure-theory/lebesgue-integral
6. σ-algebra
7. Generated σ-algebra
8. Borel algebra
9. Measurable function
10. Generators of the real Borel algebra (incomplete)
11. Measure
12. σ-algebra is closed under countable intersections
13. Group
14. Ring
15. Field
16. Vector Space
17. Probability
18. Random variable
19. Expected value of a random variable
20. Linearity of expectation
21. Covariance of 2 random variables
22. Linearity of expectation for matrices