math111_logo Partitioned Matrix


Exercise Let A be an invertible matrix. Find X, Y, S, such that

[ A B ] = [ I O ] [ A O ] [ I Y ]
C D X I O S O I

Then find out the condition for the matrix to be invertible.

Answer

Exercise A block upper triangular matrix is a partitioned matrix similar to the usual upper triangular matrix, except the entries are matrices instead of numbers, and the diagonal blocks are square matrices.

[ A1 # . . # ], A1, A2, ..., Ak are square matrices.
O A2 . . #
: :   :
O O . . Ak

What does the multiplication of two block upper triangular matrices look like? What about the inverse of a block upper triangular matrix? What about the similar problems for block lower triangular matrix and block diagonal matrix.

Answer