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# 4. Splitting criteria

(4.1) Next we want to characterize those row-minimal matrices belonging to a decomposable module. The idea comes from the fact that the canonical homomorphism factors via M and that the induced surjection is an isomorphism. Hence, it reflects a direct sum decomposition of M if and only if Q is equivalent to a block diagonal matrix Q0. Then Q is necessarily row-minimal with respect to J0 and J'0.

Proposition 6
If Q is row-minimal, then is decomposable if and only if for some permutation .

Assume M splits. Then we know and, for some permutation , ; i.e., and By Proposition 5(i2), we obtain

The same holds for J'0 , because Q is J'0-row-minimal, too; that is,

But then the composition of the canonical surjection

with

must be an isomorphism, showing .
Note: and are computed by a standard basis computation with respect to a certain module ordering in the components. So we simply have to check whether

has a solution X, which corresponds to a simple syzygy computation.

(4.2) Before computing a solution X, it is necessary for Q to be row-minimal, which may be tested using methods discussed in the next section. Usually Q is not row-minimal, in which case we shall use an algorithm to transform Q into an equivalent and J0-row-minimal matrix. But we do not know a procedure to obtain row-minimality for all subsets immediately. Hence we shall formulate a second version of the splitting criteria.

Proposition 7
If Q is J0-row-minimal, then

By Proposition 5, it follows that

then and . Therefore

and , giving

from which it may be concluded that

The other direction is obvious. Note:
if and only if C' - VA' = D'V' for some V'; i.e. we obtain

Corollary 8
If Q is J0-row-minimal, then for an ordered standard basis of Q we have:
if and only if the equation VA'+D' V'=C' has a solution (V, V').

Deciding the solvability of the above equation is reduced to a lifting computation: Let and be matrices corresponding to and , and let [V], [V'], [C'] denote flattenings to a column vector of V, V', C', respectively. Then we obtain

that is, [C'] can be lifted to if and only if the equation has a solution.

Next: 5. Criteria for row-minimality Up: Splitting algorithm for vector Previous: 3. Row-minimal matrices
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