Schur transformation

Direct comparisons can be made with the Schur transformation in the linear case. Other techniques based on the use of algebraic sets for the approximation of non-linear inertial manifolds have also been developed [192] but have not yet been applied in combustion systems. 

Discretization. The above DAE can, in principle, be discretized by any suitable method, given for instance in textbooks like [12] or [3]. For ease of presentation, we here concentrate on the rather basic linearly implicit Euler discretization. When applied to the separated form (2.5), which requires the block-Schur transformation first, this discretization reads ... 

Proof. By Proposition 3.5, since V2 is finite dimensional we know that there is an orthogonal projection 112 with range V2. Because p is unitary, the linear transformation 112 is a homomorphism of representations by Proposition 5.4. Thus by Exercise 5.15 the restriction of 112 to Vi is a homomorphism of representations. By hypothesis, this homomorphism cannot be injective. Hence Schur s lemma (Proposition 6.2) implies that since Vi is irreducible, fl2[Vi] is the trivial subspace. In other words, Vi is perpendicidar to V2. ... 

We can apply Schur s lemma (Proposition 6.2) to see that for any function a e Z the linear transformation 7), is a constant multiple of Ta . Consider the linear transformation r o Tan Y By Proposition 6.1, this... 

In this case, the transformation of 11 to T has brought 1 (R) into block-diagonal form and the matrix representation T was therefore reducible. But if T is irreducible, then dk = dh V k, and D is a constant matrix, that is the constant d, times the unit matrix. But if UHU- 1 is a multiple of the unit matrix, then so is H. And if H i and H2 are multiples of the unit matrix, then so also is M = VifH, — iH2), which proves Schur s lemma. 

We shall also need the following corollary to Schur s lemma the complete set of matrices Z that transforms F7 into F is cZ, where Z is one such matrix and c / 0 if F and F7 are unitary then Z contains unitary matrices and the complete set of unitary matrices that transform F7 into F is c 7Z. ... 

Z.A. Matysina and D.V. Schur, Hydrogen and solid phase transformations in metals, alloys andfullerites (in Russian). (Izd. Nauka i obrazovanie, Dnepropetrovsk, 2002). 

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