Assume A - SI_n 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 17.

Data from in Exercise 17

Suppose A₁₁ is invertible. Find X and Y such that

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.

Transcribed Image Text:

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