Believe it or not, here is a social ill we cannot blame on the hip-hop. No, this one falls squarely in the lap of the other evil of our times: mathematicians. Specifically: Emmy Noether, who was probably Emmy Nöther before relocating to Pennsyltucky from Germany, a juicy tidbit suspiciously omitted from her Wikipedia bio. Still, an immigrant conformity eminently understandable. Having spent time in Appalachia, I can state with some confidence they don't much cotton to the diacritically challenged in them parts. It didn't take a hole in the ground to put the bottom in their face, as the Cooleys wrote (albeit of their beloved Alabama).
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| Word Eigenvalue Day is Feb 14th! |
Noether's [sic] Theorem states that every conservation principle (momentum, energy, etc) can be expressed as a symmetry. Of the Lagrangian, although from here you're on your own. The concept sounds straightforward enough, but the details get all quantum mechanic-y and quantum mechanics makes about as much sense to LabKitty as a Mozart opera.
Whether this is responsible for the manifest symmetries manifest in professional matrix applications is hardly the point. It provides segue to the topic at hand, which for LabKitty is the usual Litmus test deciding if a factoid makes it into the post. What does quantum mechanics have to do with the real world, you stupid cat? you may be shouting right now. Well, it does for the purposes of today's slurred ranting. And while slurring usually happens in the mouth part and not the fingers, I thought we agreed you'd stop judging.
Real-world applications come from physics not from some math major drum circle, is the point. That pedigree has far-reaching consequences. Look no further than finite element and all its kittens. FEM is a $100 billion industry in Rhode Island alone, employing a literal army of engineering Ph.D.s who had hoped to get an academic position until the promise of fat cash and a company car led them from their tru calling. And finite element generates nothing but symmetric matrices all the live long day.
The Eigenvalue Problem
The problem goes as follows: Given an nxn matrix A, we seek an nx1 (column) vector x and a scalar λ such that Ax = λx. You may recall various methods for finding x and λ, either by hand or by computation, and more to the point several x and their associated λ, as an nxn matrix can have as many as n of each.
If we take the transpose, we obtain xTAT = λxT which is also perfectly valid. The vector x now appears as an 1xn (row) vector. You have to switch the order of the multiplication on the LHS but not on the right, for on the right the transpose of a scalar is just the scalar, and it doesn't matter whether we write λxT or xTλ.
Let's define y = xT, for comparison. Compare
Ax = λx
and
y AT = λy
A matrix and its transpose have the same eigenvalues. However, the eigenvector x and the eigenvector y will only be the same if A is symmetric (by "same" I mean "have the same entries" -- clearly one is a column vector and the other is a row vector).
Hence the lesson: An unsymmetric matrix does not have "eigenvectors," it has "left eigenvectors" and "right eigenvectors."
If you are, e.g., a finite element jock, this may be news to you. Sure, a professor may have mentioned this quirk somewhere in your travels and there it remained, the interim workload burying such unpleasantries like a LabKitty family secret about the mercousins. Fact turned to myth, myth turned to legend. And some things that should not have been forgotten, were.
Yet, a pressing question arises: If symmetry comes from physics, how is apostasy even possible? What sort of monsters traffic in unsymmetric matrices? How can they force themselves into our plane of existence? And how can they be identified and shunned so you don't look like an ass at a campus mixer, job interview, or thesis defense?
Squishy
Alas, matrix applications are not all symmetric rainbows and unicorns. We need look no further than the life around us.
It is 1945. Ecology is a-titter. P.H. Leslie, working at the Bureau of Animal Population at Oxford, has just published his masterpiece: On the use of matrices in certain population mathematics. In it, he generalizes the concept of growth rate for application to an age-structured population. Consider a bacterium. It divides to become two, then four, and so on to infinity in geometric glee. But more complex organisms exhibit a life cycle. The newborn must age before reproducing in turn. Fecundity and survival are age dependent. How to reduce this life's rich bounty to cold unfeeling mathematics?
Footnote: By "a-titter" I mean ecologists all but ignored Leslie's work. Although today there are textbooks on matrix population modeling thick enough to stun a Mersey (that's a type of cow, for you city folk), at mid-century matrices were viewed as some sort of impenetrable exotica. Heisenberg's (matrix) version of quantum mechanics was eclipsed by Schroedinger's formulation arguably for this reason (although Werner's Nazi sympathies didn't help). I must ask: just what the heck did people study for a STEM degree back then? These days, you can't get past first semester without an understanding of matrix algebra. But I digress.
Leslie classified organisms by age. Thus, a population is not quantified by a single number denoting its size, but rather by a vector whose entries denote the number (or, sometimes, density) of individuals in each age class. The category resolution is typically the breeding period, e.g., one year for annual organisms. This works best for semelparous species like salmon or undergraduates, but a Leslie matrix model can be attempted for continuously-breeding organisms at a cost of more effort and less accuracy.
Footnote: if fine-scale resolution is not necessary for the issue at hand (nor justified by the field data, which are notoriously labor-intensive in population ecology) you may lump your individuals into simplifying albeit still-meaningful categories, say, newborn, juvenile, adult, and post-reproductive. A model of academics might comprise categories: postdoc, assistant professor, tenured, and post-productive. Further simplification is possible by merging the final two categories without much loss of accuracy.
To simulate the population dynamics, we multiply our population vector -- call it "n" -- by a matrix A, the product giving us the population vector at the next time step. Obviously, a population changes in two ways: individuals age and individuals produce babies. Recalling the rules of matrix-vector multiplication, this demands a matrix exhibiting the following structure
Here, then, a Leslie matrix. The P's are the probability of surviving from one year to the next; the F's are the fecundity of each age group. These come from a life history table, which is in turn constructed from field observations. Many, many, details are being glossed over here. Not to mention that, in the three-score interim, Leslie's simple idea has been flogged to create complexities he never dared dream, for that is what you do if you want tenure. See Caswell's weighty tome for the current state of the art.
Footnote: Interestingly, a Leslie model only includes females, as the contribution of males to population dynamics is considered negligible. Cue wingnut radio bloviating about feminazis taking over the universities.
Punchline
Steering the grand yacht of LabKitty back into the wind of today's topic, note the Leslie matrix is not symmetric. Hence, unlike its better-groomed cousins which pervade applied mathematics, a Leslie matrix has both left and right eigenvectors. Do these have a physical interpretation? Of course they do. (Cut open any problem in applied mathematics and it will bleed eigenvectors, a professor once said.)
Just as the dynamics of a population consisting of a single age group can be characterized by a single number -- it's growth rate -- the dynamics of an age-structured population are determined by the dominant eigenvalue of its Leslie matrix. The population grows if this is greater than one, shrinks if less than one, and oscillates if complex. The associated right eigenvector is the stable age distribution. That is, the population may be growing or shrinking, but the proportion of the population in each age class will become constant. If you are familiar with vibration analysis, this may remind you of the frequency / mode shape information provided by eigenvalue/vector pairs of a structure's stiffness matrix.
The left eigenvector of a Leslie matrix gives the reproductive value of the various age classes. Put simply, it tells you how many offspring, on average, each class leaves behind. This is useful for population control, as it identifies which age class will give the most bang for the conservation (or extermination) buck.
Again the lesson: An unsymmetric matrix does not have "eigenvectors," it has "left eigenvectors" and "right eigenvectors." The two are different, and have different physical interpretations.
Footnote: Computationally, you can obtain a left eigenvector by feeding the transpose of a matrix to your eigendoodle software (which is probably MATLAB, unless you're an Octave hippy. And, yes, I know "MATLAB" is supposed to be written as small caps. Blogger again stabs LabKitty in the back. Eu tu, Google?)
Epilogue
What have we learned today? Not much. We learned that trying to format a matrix in HTML is a fool's errand. We learned that you can type many mathematics words after surprisingly many beers. And we learned that you can type many mathematics words after surprisingly many beers.
But like the Psalms or the works of the Roman playwright Terence, we are obligated to provide a moral, even if the work itself is simply intended for spiritual reflection or a distraction from the unchlorinated aqueduct. The moral is: Ecologists are people, too. Even if nobody is paying them a bazillion dollars to make a finite element model of an SR-71 airframe, or the global effects of an EMP detonation over Honolulu, or the alteration of S-wave propagation caused by North Korean tunnels under the DMZ, ecologists also have computational needs. Needs that are ill-met should you ignore their unsymmetricness. Their matrices cannot be tackled by skyline or half-bandwidth solvers, nor is there any guarantee they can be diagonalized. Their eigenvalues may well be defective, leading us to generalized eigenvectors and the Jordan form. The Gog and Magog of matrix algebra. Hardships never faced by you symmetric dandies, as you sit and sip your spice wine and are attended by lithe and sultry attendants wearing only the finest of silk.
And, of course, double the eigengoodness. Not just the Leslie matrix, but all who look into the funhouse mirror only to find their transposed countenance altered. An unsymmetric matrix does not have "eigenvectors," it has "left eigenvectors" and "right eigenvectors," a wise person once wrote. You, like Peter Parker, may choose to view this as gift or curse.
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