Assume A - sIn is invertible and view (8) as a system of two matrix equations. Solve the top equation for x and substitute into the bottom equation. The result is an equation of the form W (s) u = y, where W (s) is a matrix that depends on s. W (s) is called the transfer function of the system because it transforms the input u into the output y. Find W (s) and describe how it is related to the partitioned system matrix on the left side of (8). See Exercise 16.
Exercise 16
Let A =
If A11 is invertible, then the matrix S = A22 - A21-1 A11 A12 is called the Schur complement of A11. Likewise, if A22 is invertible, the matrix A11 A12 - A-122 A21 is called the Schur complement of A22. Suppose A11 is invertible. Find X and Y such that
