Exercise Let A be an invertible matrix. Find X, Y, S, such that
|[||A||B||] = [||I||O||] [||A||O||] [||I||Y||]|
Then find out the condition for the matrix to be invertible.
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.|
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.