Suppose the transfer function W(s) in Exercise 21 is invertible for some s. It can be shown

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Suppose the transfer function W(s) in Exercise 21 is invertible for some s. It can be shown that the inverse transfer function W (s)-1, which transforms outputs into inputs, is the
Schur complement of A - BC - sIn for the matrix below. Find this Schur complement. See Exercise 17.


image



Data from in Exercise 21


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).



Data from in Exercise 17


Suppose A11 is invertible. Find X and Y such thatimage


where S = A22 - A21 A-¹11 A12. The matrix S is called the Schur complement of A11. Likewise, if A22 is invertible, the matrix A11 - A12 4-122 421 is called the Schur complement of A22. Such expressions occur frequently in the theory of systems engineering, and elsewhere.

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Linear Algebra And Its Applications

ISBN: 9781292351216

6th Global Edition

Authors: David Lay, Steven Lay, Judi McDonald

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