16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 7.4.5

The Rational and Jordan Forms

Mathematics — Linear Algebra · invariant-factors-jordan-form

Problem

Let TT be the linear operator on R8R^8 which is represented in the standard basis by the matrix A=[11111111000000010000000−10110000100011000011111010−1−1−1−101−100000000].A = \begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\ 0 & -1 & -1 & -1 & -1 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix}. (a) Find the characteristic polynomial and the invariant factors. (b) Find the primary decomposition of R8R^8 under TT and the projections on the primary components. Find cyclic decompositions of each primary component as in Theorem 3. (c) Find the Jordan form of AA. (d) Find a direct-sum decomposition of R8R^8 into TT-cyclic subspaces as in Theorem 3. (Hint: One way to do this is to use the results in (b) and an appropriate generalization of the ideas discussed in Example 4.)

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