The Shape of Love

Happy World Eigenvalue Day! An annual tradition still celebrated in certain circles, albeit the radius contracting with each passing year. It was just a dream some of us had, Joni Mitchell once sang, a sentiment that applies to much more than eigenvalues in Year of our Lord 2026 but I digress.

If you'd like the full WED story, here's the original skinny; a site seemingly cursed with more popups and fewer working links each time I visit. (Someday I'll move the page to LabKitty actual, assuming that wouldn't screw the SEO pooches.)

But Eigenvalue Day it remains and the Nerd Code obliges me to regale you with an eigenvalue-themed post. So here it is.

Read on.

Happy Eigenvalue Day!

World Eigenvalue Day 2021

Good morrow cousins!

And it is a good morrow indeed, for it is World Eigenvalue Day. A perennial Valentine's Day alternative here at the LabKitty that hasn't quite yet caught on in the "real world" but no matter. WED commands a small but festive band of devotees. And why should it not? Self-love is the purest form of love, as they say, because before you can stop hating others you must learn to stop hating yourself.

There are also usually cupcakes.

In additional to the traditional WED gift-giving traditions, each year I offer a small tidbit of eigenvalue goodness, be it a Javascript for displaying Gershgorin circles, a Javascript for computing all the things, a list of eigenfactoids, a creative reinterpretation of eigenlore, or a creative reinterpretation of eigenlore. The more adventurous / confused might enjoy the original WED announcement which goes back to an early LabKitty flirtation with Hubpages. (Yeah, as if you don't have any youthful indiscretions hiding in your closet, like sophomore year when you wore black eyeliner and listened to the Smiths. If we are to grow as beings of pure Internet, we must own our mistakes.)

See also: gift-giving.

Each year we push ever deeper into the fetid swamp that is mathematics to recover a gem and this year is no different. I can only imagine the tenterhooks on which you must be perched, waiting for the eigenmanna to gush forth like young tikes on Christmas 'morn.

Well, as Kirk said to Khan, here it comes.

Let's Make Gershgorin Circles!

Of all the gaps matriculating from Circle Pines U left in my psyche -- how to knot a cravat, the difference between good flan and bad flan, world history before the 17th century or US after the 19th -- failure to even mention Gershgorin circles stands as a particularly egregious transgression.

In sophomore year we deconstructed Descartes' Meditations, this in the guise of a philosophy graduate student opining on the tangible gustatory effects of imagined peanut butter sandwiches. Perhaps it was a metaphor for sexual arousal, which would have made for more spirited recitation albeit risking pointed inquiry from the CP Chamber of Commerce, who already did not much care for our fancy book lernin'. Alas, whatever his intent, it was lost on us.

In junior year we discussed Lucifer State, a novel of dystopian future I rather enjoyed but is no longer in print, or so the Internet tells me. (Such not being the damning praise it once was, what with the Amazon self-publication machine Heutzutage delivering any provocatively-titled finger barfing -- Moist Vibrations, Debbie Does Due Diligence, Triggered -- to the tenterhooked browser.) Mostly I remember the prof lectured wearing sunglasses, which on more than one occasion concealed a black eye.

Then in senior year there was Herman Melville and the Forms, which was a comparative literature course. Unless that was freshman year, then it was a punk band. I think. TBH, the whole experience is something of a blur. I took classes in economics and FORTRAN, racquetball and contract law, geology and differential equations. I'm not entirely sure what my major was.

Someone could have devoted a lecture to Gershgorin circles, is my point. As I hope I have convinced you, the remainder of my education would not have suffered.

Besides, Gershgorin circles are muy cool.

Eigenvalue Day 2019 -- The Twelve Days of Eigenvalues

When LabKitty t'was but a kitten, our favorite winter holiday tradition was the Twelve Days Of Christmas. Known to most as the song (and maybe one of those uncanny valley stop animations), the LK family spin on the classic took the form of gift giving, as things in the consumerism-crazed '80s often did.

In detail, mom would hang our Twelve Days of Christmas Gift-O-Rama on the wall, a discarded turnout blanket onto which she (actually, my crazy aunt now that I think of it) had sewn a staggered row of twelve festive numbered pockets and into each of which was tucked a wrapped present. These were opened one per morning, in order, starting on December 12 (because 12 + 12 = 25. Dur-hey).

The gifts were simple gewgaws or candy, for it was an innocent time, years before the throes of adolescence would demand more demanding fare such as xboxes and carcanos (which did not fit into the wee pockets). But me and my sister loved the heck out of the TDoC while it lasted, if for no other reason than trying to guess tomorrow's gift via surreptitious wrapping paper squeezings when mom was out of earshot.

It's the sort of folksy tale Sarah Vowell or David Sedaris might relate on NPR. However, we here at the LabKitty are academics and academics are contractually obligated to crush the fun out of childhood memories, be they jump rope atrocities, baby doll penises, or, in this case, sing-song affirmation of the corrupt marriage between late-stage capitalism and a cherry-picked Judeo-Christian value system.

In that spirit, I give you The Twelve Days of Eigenvalues.

Eigenpotato

Synopsis: Titled to be maximally search engine hostile, today's post is a whimsical look at computing eigenvalues, and computing eigenvalues in Javascript, and gives you the code for free. Eigenvalues!

LabKitty loves eigenvalues. That's fairly well established in both the Internet and squishy realms. Odd, then, that I suck at computing eigenvalues. Sure, I can work a 2x2 like Sunday morning, but beyond that I am hopeless. Finding roots of a polynomial (read: characteristic equation) is my bête noire. Quintics are of course right out. I'm told there exist formulas for quartics and cubics, but my brain has rejected that proposal like a failed organ transplant. Évariste Galois died for my sins.

Thus I turn to gronkulation, only to immediately confront a different breed of horror. My off-the-shelf computing skills are able to extract precisely one eigenvalue, and it better be real and dominant and algebraic multiplicity one. Yes, the famous Rayleigh algorithm (whom I am always confusing with the guy in the can. Prince Albert. That's a different guy, correct?). Literally beating a dead horse until it coughs up the goodies, and I cannot lie: It has a kind of appeal.

All this rather takes the fun out of eigenvalues. Bored at the party, waiting at the checkout, working double time in the seduction line, thoughts turn to eigenthoughts. But confined to a 2x2 prison I am, which come to think of it describes my experience with off-campus housing. Yes, I could model stage-structured growth or that quantum system, attractive stranger, if you would permit me to return to my office and release the Kracken ("Kracken" === R2016b running on a Darwin multicore). But if you are expecting results here and now, you, my friend, are living a lie.

Which brings us to Press.

Sanity Check: Wherefore Art Thou Similarity Transformation?

Sanity Checks are missives on a specific math point in need of clarification. I try to do so using the fewest words possible. Usually, this is still quite a few words.

Remember matrix-vector multiplication? We multiply a vector (call it x) by a matrix (call it A) and obtain a new vector: y = Ax. Somewhere in your travels you were told to stop thinking of this as matrix algebra and start thinking of it as a mapping. A is a matrix, yes, but it represents a linear mapping that operates on x and sends it somewhere, and we call the new thingy y.

Perhaps here you say da doi, of course a matrix vector product is a linear transformation and a linear transformation is a matrix vector product. But it's a bona fide theorem. Poke around a linear algebra textbook and you'll find something like: A transformation F: Rⁿ → R^m is linear if and only if it is a matrix transformation. Can you prove it? I can't, but somebody can.

Today, however, we are interested in coordinate transformations. Suppose we have a mapping of the vector x into y represented by the matrix A. That is, y = Ax. Now, suppose we change the basis. In the new basis, x → x' and y → y'. Is it still true that y' = Ax'?

No, it is not.

Happy Eigenday 2018!

Once upon a time there was a village, and next to the village was a river. The river did not flow from north to south nor east to west. Neither did it flow from south to north nor west to east. It just flowed the way it did and the villagers never gave it a moment's thought. This because all roads in the village lay either parallel to the river or at right angles to it. If a stranger appeared, say, asking directions to the bagely, a villager might respond thusly: "Good Sir, you walk toward the river three blocks, along the river four blocks, then away from the river for a block." Here in the telling the villager would pause, then hook a thumb though the button-hole in his hair shirt and look wistfully off into the distance. "There you may procure the finest bagels in all the land," he would proudly beam.

Happy Eigenday!

World Eigenvalue Day
by LabKitty

EIGENBORN -- An Algebraist's Tale

Each year on the day after Dead Dragons Day, a handful of Algebraists would gather at a secret place known only to the Brotherhood. Each year fewer came than the year before. And each year, there would be more than the year that followed. They came to see old friends, came to reminisce. They came to pay their respects to forgotten gods and lost colleagues.

They came to remember the Before Times.

On a circle of FireWolf pelts they sat and feasted. Over the years, fondue had become cheese rounds had become salty bread. And over the years, wine had become mead had become potions of Resist Sleep. They would gaze long into the fire and talk and sing their songs. Some clutched an amulet worn smooth with age, the engraving no longer legible. Most clutched only their fading memories. When the moon was directly overhead, they would toast their gods. Wronskian. Hermetian. Toeplitz.

But especially Vandermonde.

The eldest would stand. He would draw a circle in the dirt with his staff. And then, standing in the circle, he would recount the tale.

The Tale of the Eigenborn.