A partitioned matrix is a matrix that is interpreted as having been separated into sections, each containing a smaller submatrix. For example, the matrix
\[\A = \begin{bmatrix} 1 & 2 & 2 & 7 \\ 1 & 5 & 6 & 2 \\ 3 & 3 & 4 & 5 \\ 3 & 3 & 6 & 7 \end{bmatrix}\]can be partitioned into four 2×2 blocks
\[\A = \begin{bmatrix} \A_{11} & \A_{12} \\ \A_{21} & \A_{22} \end{bmatrix},\]where
\[\A_{11} = \begin{bmatrix} 1 & 2 \\ 1 & 5 \end{bmatrix},\quad \A_{12} = \begin{bmatrix} 2 & 7\\ 6 & 2 \end{bmatrix},\quad \A_{21} = \begin{bmatrix} 3 & 3 \\ 3 & 3 \end{bmatrix},\quad \A_{22} = \begin{bmatrix} 4 & 5 \\ 6 & 7 \end{bmatrix}.\]This is a useful way of organizing a matrix that comes up in several contexts, such as finding inverses, finding marginal and conditional distributions, and separating the information matrix into parameters of interest and nuisance parameters (which has implications for the CRLB).
Symmetric matrices
Note that in the special case of a symmetric matrix, \(\A_{12} = \A_{21} \Tr\).
Transpose
The transpose of a partitioned matrix is
\[\A \Tr = \begin{bmatrix} \A_{11} \Tr & \A_{21} \Tr \\ \A_{12} \Tr & \A_{22} \Tr \end{bmatrix};\]Note that we have taken the transpose of each submatrix, but also switched the matrices on the off-diagonals.