Stat 5101 Lecture Notes - School of Statistics

5.2. THE MULTIVARIATE NORMAL DISTRIBUTION 147indicates a partitioned matrix. Here each of the B ij is itself a matrix. B is justthe matrix having the elements of B 11 in its upper left corner, with the elementsof B 12 to their right, and so forth. Of course the dimensions of the B ij must fittogether the right way.One thing about partitioned matrices that makes them very useful is thatmatrix multiplication looks “just like” matrix multiplication of non-partitionedmatrices. You just treat the matrices like scalar elements of an ordinary array( )( )B11 B 12 C11 C 12=B 21 B 22 C 21 C 22B 21( )B11 C 11 + B 12 C 21 B 11 C 12 + B 12 C 22B 21 C 11 + B 22 C 21 B 21 C 12 + B 22 C 22If one of the matrixes is a partitioned column vector, it looks like the multiplicationof a vector by a matrix( )( ) ( )B11 B 12 x1 B11 x= 1 + B 12 x 2B 22 x 2 B 21 x 1 + B 22 x 2and similarly for(x1) ′ ( )(B11 B 12 x1)= ( xx 2 B 21 B 22 x ′ 1 x ′ 22) ( )( )B 11 B 12 x1B 21 B 22 x 2= ( ) ( )x ′ 1 x ′ B 11 x 1 + B 12 x 22B 21 x 1 + B 22 x 2= x ′ 1B 11 x 1 + x ′ 1B 12 x 2 + x ′ 2B 21 x 1 + x ′ 2B 22 x 2Of course, in all of these, the dimensions have be such that the matrix multiplicationsmake sense.If X is a partitioned random vector( )X1X = , (5.30a)X 2then its mean mean vector iswhereand its variance matrix iswhereM =µ =(µ1µ 2), (5.30b)µ i = E(X i ),( )M11 M 12, (5.30c)M 21 M 22M ij =cov(X i ,X j ).Again, every thing looks very analogous to the situation with scalar rather thanvector or matrix components.