C D A B in .NET Writer data matrix barcodes in .NET C D A B

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C D A B generate, create data matrix barcode none on .net projects ISO Standards Overview Realizations With the nota tion introduced in part (iii) of Lemma 9.5.7, we have (since is input normalized and is output normalized; hence C C = 1 and BB = 1) BB = C C = E .

X , Thus, BB = F, C C = F X .. is (Hankel) b Data Matrix for .NET alanced. By Theorem 9.

5.3(i),. . A sim ilar computation shows that B B = . hence CC = That a balanced realization is unique up to a unitary similarity transformation in its state space follows from the nal claim in Theorem 9.

2.5 and the fact is independent of the realization (as long as it is balanced). that CC = That CC.

N (C ) = . N ( ) and C C = BB N ( ) , B B N (B) = are unitarily similar to each other follows from Lemma A.2.5.

In the proof of Lemma 9.5.7 we use the following fundamental lemma.

Lemma 9.5.8 Let U , Y , and W X V be Hilbert spaces, where the embeddings are continuous and dense, let E B(V ) be injective, self-adjoint (with respect to the inner product in V ), and suppose that E maps V isometrically onto X and that E .

X maps X isom Data Matrix ECC200 for .NET etrically onto W . Let A be a contraction on V , and suppose that W is invariant under A, and that A := A.

W is a contraction on W . Then A := A X is a contra Data Matrix barcode for .NET ction on X (in particular, A maps X into X ). Proof Let A B(V ) and A B(W ) be the adjoints of A, respectively A, when we identify the dual of W with V (with X as pivot space), let A B(W ) be the adjoint of A with respect to the inner product in W , and let A B(V ) be the adjoint of A with respect to the inner product in V .

All of these operators are contractions in the indicated spaces, A = A , and by Proposition 3.6.2, .

W A = E 1 A E,. = CC = CEC = CC CC ,. A = E A E 1 ,. A = E 2 A E 2 . De ne B := A A = A A. Then B B(W ), and B is positive with respect to the inner product in X on W since, for all w W , w, Bw. = w, E Bw = Aw, Aw = w, A E Aw 0. = Aw, E Aw This implies that B can be extended to a (possibly unbounded) positive selfadjoint operator (with respect to the inner product in X ) mapping D (B) X into X (see, e.g., Kato 1980, Theorem 3.

4, p. 268, Corollary 1.28, p.

318, and Theorem 2.6, p. 323).

We still denote the extended operator by the same letter B. Note that W D (B n ) for all n = 1, 2, 3, . .

. since B was originally de ned. 9.5 Normalized and balanced realizations on W and B ma ps W into itself. The operator B has a spectral resolution F( d ), and for all n = 1, 2, 3, . .

. and x D (B n ) (in particular, for all x W ) Bn x =. n F( d )x. Let be the orthogonal projection x = F((1, )) = (1, ) F( d )x, let X 1 = N ( ) and X 2 = R ( ), and let B1 = B(1 ), B2 = B . Then commutes n with B, B1 is a contraction mapping X into X 1 , D B2 = D (B n ) W for all n = 1, 2, 3, . .

., and for all x W and n = 1, 2, 3, . .

. , x, B n x.
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