By Kleshchev A.S.

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**Additional resources for 1-Cohomologies of a special linear group with coefficients in a module of truncated polynomials**

**Example text**

4870 . Find the inverse of A and also IIAII, IIA-III for the matrix norms indu-ced by the Chebyshev and Manhattan vector norms. 3. 4. 3 is -6] 25 -41 10 W-I= [ -41 68 -17 10. 10 -17 5-3 -6 10 -3 2 Find the matrix norms of W, W-I induced by the Chebyshev and Manhattan vector norms. 5. Express the matrix H4=A13 as a product LDL' where the diagonal elements of L are all 1. '. ) 49 The Inversion Problem il: Practical Computation Find the matrix norms of H 4 , Hi! induced by the Chebyshev and Manhattan vector norms.

X* x = 1, and if Xl ~ -1 is real then the matrix [ Xl U= i _I + (1 i* ] + Xl) -1 i i* ' which has x as its first column, is unitary. 25. Describe how to obtain the reduced row-echelon form of a matrix. What difficulties can arise if one attempts to implement this reduction algorithm on a computer? 26. Suppose the n-dimensional column vectors Xl' X 2 , ... , Xr are linearly independent. Describe how to obtain n-r vectors Xr+l, X r +2, ... , Xn so that Xl' ... , Xn forms a basis for Rn. The Inversion Problem I: Theoretical Arithmetic 43 Discuss the implementation of this process on a computer.

7. 1 is available the solution of the system Ax=b is easy. We have LUx=b which can be written as Ux=y, Ly=b. So, x is obtained by solving a triangular system after obtaining the righthand side y by the solution of another triangular system. As we have seen, the solution of such systems involve about n2 j2 multiplications. It is also clear that det A = det L U = det L det U = II Ii i II Ui i and so det A can be evaluated by at most 2n additional multiplications. What we have to do now is to find the extent of the computation of L, U.

### 1-Cohomologies of a special linear group with coefficients in a module of truncated polynomials by Kleshchev A.S.

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