# Ljung–Box test for a multivariate time series?

From Tsay's Analysis of Financial Time Series,

For a univariate weakly stationary time series $r_t$, its sample autocorrelation function $\hat{\rho}_l$ is defined as:

and the Ljung-Box test is

For a weakly stationary multivariate time series $r_t$, the sample lag-$l$ cross-covariance matrix $\hat{\Gamma}_l$ and sample cross-correlation matrix $\hat{\rho}_l$ of $r_t$ is defined as

The multivariate Ljung-Box test is:

I wonder how to see the test statistic for a multivariate time series is a generalization of the one for a univariate time series.

• Why do the coefficients $T(T+2)$ become $T^2$?
• Why does $\hat{\rho}_l^2$ become the trace of something?
• How is the representation in Kronecker product equivalent to (8.7)?