Theorem (Finite Variance = Autocorrelation symmetric + positive semidefinite)

Let $\mathbf{X}=(X_{1},\dots,X_{k})^{T}$ be a random vector having finite variance. Then the $k\times k$ autocorrelation matrix $\mathbf{R_{X}}=\{ E[X_{i}X_{j}] \}$ is symmetric and positive semidefinite.

Theorem (Linear transform for autocorrelation matrices)

If $\mathbf{X}=(X_{1},\dots,X_{k})^{T}$, $\mathbf{A}$ is a $k\times k$ matrix, and $\mathbf{Y}=\mathbf{AX}$, then $$\mathbf{R_{Y}}=\mathbf{AR_{X}A}^{T}$$

Theorem (Determinants for linearly transformed autocorrelation matrices)

If $\mathbf{Y}=\mathbf{AX}$ then $$\det \mathbf{R_{Y}}=\det \mathbf{A}^{2}\det \mathbf{R_{X}}$$If $\mathbf{A}$ is orthogonal then $$\det \mathbf{R_{Y}}=\det \mathbf{R_{X}}$$