By Bart H., Gohberg I., Kaashoek M. A.
The current ebook offers with factorization difficulties for matrix and operator features. the issues originate from, or are inspired via, the speculation of non-selfadjoint operators, the idea of matrix polynomials, mathematical structures and keep an eye on concept, the idea of Riccati equations, inversion of convolution operators, conception of task scheduling in operations study. The e-book systematically employs a geometrical precept of factorization which has its origins within the country area thought of linear input-output structures and within the thought of attribute operator services. This precept permits one to house diversified factorizations from one viewpoint. lined are canonical factorization, minimum and non-minimal factorizations, pseudo-canonical factorization, and diverse kinds of measure one factorization.Considerable recognition is given to the problem of balance of factorization which by way of the kingdom house strategy consists of balance of invariant subspaces.invariant subspaces.
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Extra info for Factorization of Matix and Operator Functions: The State Space Method
To prove the if part, we argue as follows. The fact that M is invariant under A implies that A21 = 0. As ⎤ ⎡ A11 − B1 D2−1 D1−1 C1 A12 − B1 D2−1 D1−1 C2 ⎥ ⎢ A× = A − BD−1 C = ⎣ ⎦ −1 −1 −1 −1 −B2 D2 D1 C1 A22 − B2 D2 D1 C1 leaves the space M × invariant, we have A12 = B1 D2−1 D1−1 C2 . But then the conclusion Θ = Θ1 Θ2 follows directly from the definition of the product of two systems. 2. 3, we consider the case when the input/output space Y is finite-dimensional. 3. Here are the details. 4. Let Θ = (A, B, C, D; X, Y ) be a biproper system, let M and M × be subspaces of X, and assume X = M ∔ M × .
Products 31 Proof. Take λ ∈ ρ(A1 ) ∩ ρ(A2 ). Then, as can be verified by direct computation, ⎡ ⎤ −1 (λ − A1 ) H(λ) ⎦ : X1 + ˙ X2 → X1 + ˙ X2 , (λ − A)−1 = ⎣ −1 0 (λ − A2 ) −1 −1 where H(λ) = − (λ − A1 ) B1 C2 (λ − A2 ) . Using this and the expressions for B, C and D given prior to the theorem, we have W (λ) = D + C(λ − A)−1 B. The right-hand side of the latter identity transforms into = D1 D2 + = ⎣ D1 C2 C1 D1 D2 + ⎡ (λ − A1 )−1 −1 (λ − A2 ) 0 C1 (λ − A1 )−1 H(λ) ⎤⎡ ⎦⎣ B1 D2 B2 C1 H(λ) + D1 C2 (λ − A2 )−1 ⎤ ⎦ B1 D2 B2 D1 + C1 (λ − A1 )−1 B1 D2 + C2 (λ − A2 )−1 B2 .
5. 6 we prove the following result. 7. Let Θ = (A, B, C; X, Cm ) be a unital system with a finite-dimensional state space X, and let W be the transfer function of Θ. Assume that A is diagonalizable. Then W admits a factorization of the following form W (λ) = (Im + 1 1 R1 ) · · · (Im + Rn ), λ − λ1 λ − λn where λ1 , . . , λn are the eigenvalues of A counted according to algebraic multiplicity, and R1 , . . , Rn are m × m matrices of rank at most one. Recall that A is called diagonalizable if A is similar to a diagonal matrix.