One mass removed, and one eigenvalue gone

A chain of eight masses joined by springs, with dampers across the same connections, does not produce an ordinary eigenvalue problem. It produces (λ²M + λC + K)x = 0, and the object with a spectrum is a matrix that depends on its own eigenvalue rather than a matrix. Its eigenvalues are the roots of det Q(λ), which for eight rows is a polynomial of degree sixteen, so an eight-by-eight problem has sixteen answers.

Now take one of the masses away. Not lighten it — remove it, so that the coordinate has stiffness and damping attached to it and no inertia at all. That is what a rigid link looks like inside a model that otherwise has dynamics, and what an algebraic constraint looks like after it has been substituted in. M becomes singular, det Q loses degree, and the eigenvalues that have gone have gone to infinity.

The question is how many are left, and there are two plausible answers.

The first is that the degree falls by one for each missing mass: 2n − k, which at eight masses with one removed is fifteen. The second is the one a reader reaches by picturing the object rather than the determinant. A mass that is not there is a node that is not there; what remains is a chain of seven; a chain of seven has fourteen eigenvalues. That reasoning is clean, physical, and wrong, and it is wrong in a way that no amount of care about the arithmetic would catch, because it is not a claim about arithmetic.

2n − k is not 2(n − k). The two counts differ by k, which is to say that every mass removed costs one eigenvalue on the first reading and two on the second, and the whole of this essay is the measurement that decides between them and the mechanism that explains why the answer is the one it is.

The count is a degree, and the degree is fifteen #

The reason this can be settled rather than argued is that the degree of det Q is available exactly.

det Q(λ) is a determinant of a matrix whose entries are quadratics, so it is a polynomial of degree at most 2n, and 2n + 1 values determine it. Evaluate it at the integers 0 through 2n, in BigInt rationals, with an exact determinant at each node, interpolate in the same rationals, and strip the leading zeros. What comes back is a list of exact coefficients, and the position of the last nonzero one is a degree.

That distinction is the point of taking this route. The other way to reach the same integer is to ask how many singular values of M are zero, and rank is a decision rather than a property: it rests on a threshold, and a threshold can be set wrongly. Here the two routes agree, because the gap the decision rests on is between a singular value of exactly zero and one of exactly one — the largest ratio between consecutive singular values is infinite, which is as easy as a rank decision gets. But they agree as two independent answers rather than as one answer computed twice, and only the first of them is a count that could not have come out otherwise. That is the same standing an eigenvalue count that cannot be slightly wrong has: an integer is right or wrong and never approximately either.

The exact route has a price, and the family here is built to pay it. Interpolation in rationals requires the entries to be integers, and a chain assembled from measured stiffnesses is not, which is why an exact answer to a measured problem refuses rather than rounds. Everything below is a statement about a chain whose masses are 0 or 1 and whose stiffness and damping are small integers, and it is exact about that chain rather than approximate about a real one.

Run it on eight masses with the first removed. The degree is fifteen, out of a possible sixteen. One eigenvalue at infinity, fifteen finite, and no fourteen anywhere in the computation.

The shortfall is the number of constraints, and it does not grow #

One measurement at one size settles nothing about which rule is being obeyed. Fifteen is 2n − k at n = 8 and k = 1, and it is also 2n − 1 for a rule that has nothing to do with k, and it is also n + 7 for a rule that has nothing to do with either. So the same computation was run across every chain length the exact interpolation can afford and every number of masses that leaves at least one behind: twenty-seven settings, from three masses to eight.

The degree is 2n − k at all twenty-seven. Not approximately, and not to a tolerance — the coefficient above the degree is exactly zero as a rational number and the coefficient at the degree is exactly not.

At one mass removed, along the sizes, the degrees are 5, 7, 9, 11, 13 and 15.

The shortfall is one at every size. It is the number of constraints and it does not scale with the problem, which is the difference between a defect and an accounting rule: a quantity that grew with n would be describing how much of the chain had been damaged, and a quantity that stays at k is describing how many equations changed order.

The other direction is the one the slider on the figure above walks. Holding the chain at eight masses and taking them away one at a time, the degrees are 15, 14, 13, 12, 11, 10 and 9 — one lost per mass, all the way down to seven of the eight coordinates carrying no inertia. At six masses the same sweep reads 11, 10, 9, 8, 7. Neither sequence bends. The deleted-node reading would have predicted 14, 12, 10, 8, 6, 4 and 2 at eight masses, and it is wrong by k at every step rather than wrong once at the end, which is what distinguishes a rule that is off by an offset from one that is off by a factor.

The extreme case is the one worth naming, because it is where the two readings are furthest apart and where the physical picture has run out entirely. Eight masses with seven removed leaves a single coordinate with inertia, and the degree is 9: two eigenvalues from the one remaining mass and one each from the seven constrained coordinates. The deleted-node reading predicts two. Seven of the nine eigenvalues belong to coordinates that reading has thrown away.

Three masses is the bottom of the range and it is worth looking at because the two candidate rules are furthest apart in proportion there. Five is 2n − k. The chain-of-two reading predicts four, which is twenty per cent of the whole count away rather than one part in sixteen, and it is not what the polynomial has. Whatever is happening is not a small correction that only shows up when the count is large.

Four masses is the direct comparison, because the opening figure is that chain with everything present. Eight eigenvalues become seven. Not six, which is what the four-becomes-three reading requires. The two figures are the same object one mass apart and the count between them moves by one.

A massless coordinate is not an absent one #

The mechanism is visible in the determinant, and it is worth stating before the measurement that confirms it, because the measurement is designed to be able to contradict it.

Every row of Q(λ) is λ² times a row of M, plus λ times a row of C, plus a row of K. A coordinate with mass contributes a row whose diagonal entry is a genuine quadratic in λ: it can supply two to the total degree. A coordinate without mass has a zero on the diagonal of M, so its diagonal entry is at most linear in λ, and the most it can supply is one. It supplies one rather than zero because the row is not empty — the coordinate is still attached to the rest of the chain, and it still has a damper.

So the accounting is per coordinate rather than per node: n − k coordinates give two each, k coordinates give one each, and the total is 2(n − k) + k, which is 2n − k. The deleted-node reading gets 2(n − k) because it assumes the missing coordinates give nothing, and the k it misses is exactly the k first-order equations that a node with no inertia still writes down.

That is also the sense in which the missing eigenvalues are constraints rather than failures. An eigenvalue with no value makes the same count for a linear pencil λA + B, where a singular A gives one infinite eigenvalue per algebraic constraint, and reaches it by a different route: there the object is a pencil and the shortfall is n − rank, here the object is a quadratic and the shortfall is measured against 2n. The two counts describe the same physical thing — a coordinate the model does not integrate — and they are not the same arithmetic, which is why the subtraction has to be done rather than translated.

The damping on the massless coordinate is the whole of it #

The paragraph above is a story, and a story about a determinant is exactly the kind of thing this collection does not publish without a way of being wrong. So it was given one.

If the extra degree comes from the massless coordinate’s own first-order term, then removing that term should cost the degree. The damping on that coordinate can be deleted — its row and column of C set to zero, leaving the stiffness untouched, so the coordinate is still coupled and still constrained and now has neither inertia nor dissipation of its own. The prediction is that the degree falls by one more per coordinate treated that way, and that a chain with all of its massless coordinates undamped finally does read 2(n − k).

It does. At six masses with two removed, the degree is 10 when both massless coordinates are damped, 9 when one of them is not, and 8 when neither is — and 8 is 2(6 − 2), the count the deleted-node reading predicted all along, now arriving at the setting that actually deletes them. At eight masses with one removed the same strip takes 15 to 14, and 14 is twice seven.

The sweep behind that is 110 settings, over every chain length from three to eight, every number of missing masses, and every number of those left undamped. The degree is 2n − k − s at all 110, where s counts the massless coordinates whose damping was removed.

Then the same measurement was pushed until it produced something the story had not predicted. Instead of clearing a coordinate’s whole row and column of C, individual entries were cleared, and the degree was compared against the rank of the damping block sitting on the massless coordinates. Across 690 such patterns the degree is 2(n − k) + rank of that block, with no exceptions.

That is sharper than the sentence it was written to test, and in one place it corrects it. At one missing mass the degree-fifteen term needs the massless coordinate’s own damper, the diagonal entry: keep only that and the degree is 15, keep only its couplings to its neighbours and the degree is 14, the same as no damping at all. The coordinate being attached to a damped chain is not enough. It has to dissipate on its own account, because the only permutation of the determinant that can reach degree 15 is the one that takes the diagonal entry out of the massless row, and every other permutation is forced to pay a second time in the column that row abandoned.

The counts are also indifferent to how hard the chain is damped, which is the other thing that had to be checked before the rank of the damping block could be called the mechanism. Twelve damping settings were run across the same chain lengths — 240 combinations — and the degree is 2n − k at every one. What matters is whether the massless block of C is singular, not how large its entries are. A count that follows from the algebra and a count that follows from the coefficients are different kinds of number, and this one is the first kind.

The fifteen are not the fourteen with one more #

A reader who has accepted that the count is fifteen still has a way to keep the picture: perhaps the fifteen are the fourteen eigenvalues of the seven-mass chain, with one extra root belonging to the constrained coordinate. That would make 2n − k a bookkeeping detail rather than a different problem.

It is not, and the test is exact. The characteristic polynomial of a genuine seven-mass chain has degree 14, as it must. Divide the massless eight-mass chain’s degree-15 polynomial by it, in rationals. If the picture were right the remainder would be zero. The remainder has degree 13.

The computed spectra say the same thing in a form that can be read. The seven-mass chain’s fourteen eigenvalues include −1 seven times and run out to −3.847759; the massless chain’s fifteen include −1 seven times and run out to −3.838855. Seven values coincide exactly and the rest do not: the worst mismatch is 0.117704, between an eigenvalue at −0.585786 on the shorter chain and the nearest thing to it at −0.703490 on the constrained one. Close enough to look like a perturbation and far enough that it is not one. Removing a mass does not leave the smaller problem alone and add a root to it; it produces a different problem of a different degree whose eigenvalues are all in slightly the wrong places.

There is a second reading of the same fact, from the route anybody actually runs. The standard reduction of the companion linearisation inverts its leading coefficient, and for the first companion form that coefficient carries M in its top-left block, so it is singular exactly when M is. The reduction cannot be formed at all. The other reduction, which inverts the trailing coefficient and returns reciprocals, does run, and it comes back with fifteen finite numbers and a sixteenth that is not a number — the reciprocal of a zero, which is what an eigenvalue at infinity looks like after the mapping. The form a real matrix can reach is doing its work correctly on both; what it cannot do is tell the caller that the sixteenth entry means the model has a constraint in it rather than that something went wrong.

Where the accounting is not the whole story #

Two things the measurement found are worth recording as limits on how far the tidy version can be taken.

The first is that the per-coordinate accounting is a statement about degrees and not about which roots appear. This family’s damping puts a root at exactly −1 for every coordinate that still has mass, and (λ + 1) divides det Q at least n − k times at all twenty-seven settings. At nine of them it divides one time more, and those nine are not scattered: every one of them has n − k equal to 2 or to 5, and a chain of two masses or of five has a stiffness eigenvalue of exactly 1, which puts a second root on top of the first. Nothing about the count changes — the degree is still 2n − k — but a reader who has learned to predict the factorisation from the number of masses will be wrong at three of the six sizes with one mass removed. The multiplicity of a root is not a rank statement, and it does not obey one.

The second is that all of this is exact about integer chains and says nothing directly about a measured one. A stiffness that arrives as 2.0000001 rather than 2 makes M nonsingular by a rounding, and the degree of det Q is then 2n with two of its coefficients of order 10⁻⁷. Nothing has been gained: the question has turned back into a rank decision, on a gap that is no longer between zero and one. That is the boundary a constraint is a weight at infinity is about from the other side, where the constraint is approached as a limit rather than imposed, and the infinite eigenvalue is a large one that has not arrived yet.

What follows for anything that counts modes #

The rule is short and its consequences are not, so they are worth going through one at a time.

A modal count is 2n − k, and a code that reports it as 2n or as 2(n − k) is wrong in opposite directions. The first over-reports by k and the second under-reports by k, and on a model with many rigid links the second is the more dangerous, because it is the one that looks physical. A structure with fifty coordinates and twelve constraints has 88 finite modes, not 100 and not 76.

A count is worth having only when the route to it cannot be talked out of an integer. The exact interpolation here is the same device that gives a count that comes out of a determinant its standing: the answer is a whole number because the computation produces whole numbers, not because a whole number was the nearest thing to what came back.

A missing mass is a modelling decision and a missing damper is a different one. The two look identical in a diagram and they differ by a degree apiece. A model that idealises a light component to massless has one arithmetic; a model that idealises it to massless and frictionless has another, and the count is the cheapest place the difference shows.

The closed form is what makes any of this checkable. Every claim above rests on a family whose answer is available independently of the computation being tested, which is the standing an answer that is known has on this site and the reason the comparison is a measurement rather than a consistency check.

And a number nobody can act on should not be reported. The count of eigenvalues at infinity is actionable: it says how many algebraic constraints the model has, and it is the same integer a singular second coefficient produced when two matrices and one problem first cost this collection an eigenvalue. The count of finite eigenvalues read off a plot, by contrast, is the number that decides nothing — it depends on where the axis was cut, and it is not a degree.