#irreducibility

Irreducible operators are very important in Mathematics. For the case of Directed Graphs, the adjacency matrix is irreducible if and only if the graph is strongly connected. Unfortunately, the canonical definition of irreducible matrices has bothered me for years. If you search for reducible on Wolfram, Wikipedia, or Planet Math you will get one of two definitions:

Definition 1: An n x n matrix A is said to be reducible if and only if for some permutation matrix P, the matrix P^T A P is block upper triangular.

or

Definition 2: A square n x n matrix A = a_ij is called reducible if the indices 1, 2, ..., n can be divided into two disjoint nonempty sets i_1, i_2, ... , i_u and j_1, j_2, ... , j_v (with u+v = n) such that a_{i_α, j_β} = 0 for α = 1, 2, ... , u and β = 1, 2, ..., v

What does this even mean? Block upper triangular? Disjoint, non-empty sets of indices? What does this have to do with reducibility? These definitions are over-complicated, overly-technical, and miss the simplistic beauty of what it means for an operator to be reducible.

What is Reducibility, Really?

As a graduate student I was tasked with extending the Perron-Frobenius Theorem to a more generalized setting. I had to think about what reducibility meant, in general. Several times my thesis supervisor hinted at such a generalization, but it never came to me intuitively until a few years ago.

Here is what the definition of a reducible matrix should be:

Definition: An N x N matrix A is reducible if there exists a non-trivial subspace invariant under A, i.e. there exists Y ⊊ ℝ_N such that A(Y) ⊆ Y.

Or, more generally for linear operators:

Definition: An linear operator T is reducible if there exists a non-trivial subspace invariant under T.

Non-Trivial Invariant Subspace?

In the above more general definition, we are saying: if T : X -> X is a linear operator, then T is reducible if there exists some non-trivial supspace Y ⊊ X such that T(Y) ⊆ Y. This means we can "reduce" the behaviour of T onto some well-defined subset Y ⊊ X. Therefore, it may be possible to analyze T in a restricted (or reduced) subspace Y. Essentially, we've "reduced" the behaviour of T to a subset of X.

Conversely, the above definition means that irreducible operators have no special behaviour for any subspace of X. That is, their behaviour cannot be further reduced; T requires the entire space X to express itself.

While I am using very loose language (what does it mean for an operator to express itself, or what is an operators "behaviour"), it is hopefully intuitive that this definition has more interpretive meaning.

Are These Definitions Equivalent?

For matrices, the invariant subspace definition and the block upper-triangular / permutation indeces definitions of reducibility are equivalent. It's a fairly straightforward proof: if a non-trivial, invariant subspace exists, then decompose your space into the direct sum of this space and its compliment. Using a permutation matrix to transform your basis, it's fairly easy to see that the matrix can be represented in block-triangular form (the proof comes down to showing that one of your blocks A_3 * Y = 0 which, since Y is non-trivial, implies the block A_3 must be zero). If your matrix is in block upper-triangular form, it's easy to see there is an invariant subspace.

Why Does This Matter?

The initial definitions are overly technical and miss the point of reducibility; that the behaviour of reducible operators can be simplified, and that irreducible operators require the entire space to express themselves. Further, the block upper-triangular and permutation indices definitions do not translate well to more generalized settings. For example, what does upper-triangular form mean in L_p spaces? It turns out there is a way to define upper-triangularity in L_p spaces (see this book), but reducibility in the context of invariant subspaces is much more intuitive.

What Next?

"Rational discourse is based less on the geometry of light than on the insistent, impenetrable density of the object, for prior to all knowledge, the source, the domain, and the boundaries of experience can be found in its dark presence. The gaze is passively linked to the primary passivity that dedicates it to the endless task of absorbing experience in its entirety, and of mastering it.... The gaze is no longer reductive, it is, rather, that which establishes the individual in his irreducible quality. And thus it becomes possible to organize a rational language around it. The object of discourse may equally well be a subject, without the figures of objectivity being in any way altered. It is this formal reorganization, in depth, rather than the abandonment of theories and old systems, that made clinical experience possible; it lifted the old Aristotelian prohibition: one could at last hold a scientifically structured discourse about an individual" (xiv).

so is the clinical experience based on a fundamental impenetrability of the object or on the assumption of its impenetrability?

i love Glissant's notion of opaquness --"that is, the irreducible density of the other"-- and it forces this consideration: what is the space, and what emerges when we theorize that bridge, between irreducibility and impenetrability? 

irreducible:  

  That cannot be reduced.   1.  a. That cannot be brought to a desired form, state, condition, etc. Const. †into, to. b. spec. That cannot be reduced to a simpler or more intelligible form; incapable of being resolved into elements, or of being brought under any recognized law or principle. ... 3. Incapable of being reduced to a smaller number or amount; the fewest or smallest possible. 4. That cannot be reduced to submission; invincible, insuperable.

i think what's cool about irreducibility when refracted through Glissant's irreducible density is that it approaches what N. says in Bedouin Hornbook, that there is an "insistent previousness evading each and every natal occasion," which is to say, the thing that avoids any moment of origin or birth; there is something previous to any sorta insistence on any point as the moment of origin. density [thickness, closeness of texture, crowded, which is to say, sociality] as irreducible means, for me, the impossibility of being exhausted or evacuated; or it approaches what i believe Heidegger argues about the piece of chalk, that it retains something of itself each time it is broken, that breaking it into halves and quarters and eighths does not reveal, nor extinguish, evacuate, exhaust the thingness of the chalk, but it remains in the pieces.

irreducibility as a concept i think calls for penetration as a means of testing, of experimentation. that is, i conceptually figure irreducibility as a solicitation, an invitation to social experimentation, a social experimentation brought to bear by Glissant.

impenetrable:

Not penetrable.

 1. That cannot be penetrated, pierced, or entered; impossible to get into or through. Const. to,by.

2. transf. and fig. Whose nature, meaning, etc. cannot be penetrated or discerned; inscrutable; unfathomable.