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.
PARTE ONE
In which we meet the hero Z'Jargo
Yes, Z'Jargo barreled well. This is fortunate for our tale, as it left him copious free time for side quests. For in addition to being a skilled craftsman, Z'Jargo was a member of the Algebraist Brotherhood, an ancient and mysterious sect schooled in mathematical majicks. These days their kind have all but vanished, lost to the dungeons of SnowFall College during the Imperial Payline Holocaust. But in the Before Times, Algebraists dispersed throughout the Realms of YonderEdge like the four conjectural winds. There they lived, learned, loved. And quested.
Be he secretive as a Duggar horde skulking in Barren Barren, you could always tell an Algebraist by the amulet he carried. On the front of the amulet was engraved a collection of runes called a ma'atrix. A ma'atrix of nine runes would be organized into three rows and three columns. A ma'atrix of sixteen into four rows and four columns. A ma'atrix has an equal number of rows and columns, the point being. There was no limit to the size of a ma'atrix, although eventually one's amulet would take on such dimensions as to resemble a gaudy trinket worn by the hop-hippers and other gibbering hooligans.
The ma'atrix an Algebraist carried was unique, both in size and in the items it contained. It was assigned to him for life upon joining the Brotherhood. Thus equipped, each new recruit was charged with a quest: go forth and discover a vecktor that is left untransmuted by thine ma'atrix, for truly then you are Eigenborn.
INTERLUDE
In which we recount Eigenlore
Vecktor transmutation was carried out at a shrine of one of the Algebraist Elder Gods. Of these there were never many, and most folk today would be hard-pressed to name even one. Wronskian the Changeling, Hermetian the Twisted. Pressetal the Snarky. An Algebraist would write a vecktor onto a scrap of parchment and place this along with his ma'atrix amulet onto the alter. Sacred words would be intoned, and thereupon the vecktor written on the parchment would change as if the very ink had become alive. The transmutation complete, the vecktor entries are changed from that which had been written.
The three-vecktor [ foo bar bletch ] might transmute to [ splat bang bang ]. Or to [ bang splat bang ]. Or to [ shriek foobie exec ]. Or to some other three-vecktor. But legend spoke of vecktors that would emerge from transmutation unchanged. As immune to ma'atrix transmutation as a dragon to fire or Imperial to reason. Following transmutation, the vecktor entries might become writ larger, or writ smaller, but they would not change, not even in their ordering.
Such a vecktor was called an eigenvecktor of the ma'atrix.
It was believed that for each ma'atrix at least one eigenvecktor existed. Yet, as we begin the telling of our tale, none had ever been discovered. The search for an eigenvecktor was the engine that drove the Brotherhood's frenzied quest. For such a finding would no doubt earn them publication, if not in Syence or Na'ature, then in one of the lessor skill-tomes such as PYAS or Acta Bork Bork.
There was one known rule: a ma'atrix would only transmute a vecktor commensurate to its size. A ma'atrix of nine entries would transmute only a vecktor with three. A ma'atrix of sixteen entries only a vector with four. And so on.
Beyond that, understanding of eigenvecktors was skimpier than the outfits in a Scarlet Blade armor mod.
PARTE THREE
In which Syd Field demands we torment the protagonist
Alas, the ma'atrix assigned to Z'Jargo comprised but four runes, arranged in two rows and two columns. This pained Z'Jargo to no end, as such a ma'atrix was but one step removed from being one rune arranged in one row and one column, which is no ma'atrix at all but just an ordinary thing. Z'Jargo could not help but envy the impressive amulets of nine or sixteen or twenty-five runes carried by Algebraists of legend. Disparage not!, Strang the Elder said to Z'Jargo one day upon noting our hero's gloomy countenance. 'Tis not the size of the ma'atrix but the magnitude of your Frobenius!, Master Strang joked, although not really in the ha-ha way.
Still, Z'Jargo was loathe to expose his amulet in the presence of others, which hindered his transmutation efforts. For an Algebraist shrine was a busy place in those days, mistaken as they often were for the shrine of a common deity. Rare was the visit not interrupted by some breed of woodland creature requiring dispatch, or some mishapen peasant just this side of the uncanny valley seeking to engage your services, or some demented adventurer stumbling forth from the underbrush trailing a limb and offering only the word troll! in a hoarse whisper before collapsing on the altar. Such abuse resulted in substantial downtime before the shrine was again usable. It wasn't like you could unhose a mystic portal by cycling the power.
Verily, Z'Jargo took these lemons life had provided and made a kick-ass half-cell, as the saying goes. In less-cryptic telling: he built his own shrine. In a back room of his hovel, and to Vandermonde the Determined, the Algebraist Deity whom he took as patron. A fine shrine to be sure, with Oakenwood inlays, picture-in-picture, and full TCP/IP back-compatibility. Even Strang the Elder had been impressed by the craftsmanship, pausing during the consecration ceremony to admire the miter work on the infant skulls Z'Jargo had affixed to the ciborium. Perhaps one day you will have my job!, Master Strang joked, although, again, not really in the ha-ha way.
Z'Jargo spent many-an-hour transmuting at his private shrine, evenings and weekends, and on holidays when the Barrerly was shut. His efforts required little field work, because a constant supply of vecktors were provided him by Algebraists passing to the nearby wenchkeep. Z'Jargo's colleagues would oft visit but never linger, surrendering a page torn from their questing journal then hurriedly excusing themselves with claims of parchments to be graded or departmental covens to be convoked. It was as if they believed their career progress would be hindered by too intimate an association. For all in the Brotherhood knew of the diminutive ma'atrix Z'Jargo carried. He'll never get tenure, they would whisper when out of earshot, and sadly shake their heads.
And so it was Z'Jargo was alone in his quest, or nearly so. His only regular companion -- an Enkidu of sorts to his Gilgamesh -- was Neuman. Neuman the Town Fool, also known as "Open-Loop Neuman," "Neuman the Potentially Stabby," and "Neudor." Neuman was a shuffler and a mumbler, with ratty clothes and wild hair that rivaled any wizard's. He had taken to skulking outside of Z'Jargo's hovel soon after the shrine to Vandermonde was complete. An apt mascot for this crazy dream, Z'Jargo thought, and perchance a good luck charm. Z'Jargo gave the fool what food he could spare. And coin too, for Neuman's begging cup was always at the ready, especially when one of the Algebraist Brothers would drop by. Not on an academic's salary!, they would demur and rush past poor Neuman, a shameful display that always left Z'Jargo with a heavy heart.
Only Strang the Elder would indulge the fool, making a point to plunk a gold piece into his cup after each visit. Neuman would clumsily retrieve the coin with his sausage fingers and test it between his green teeth, then gesture to Master Strang and mumble an inchoate greeting.
PARTE FOUR
In which we witness the birth of the Eigenborn
He had prepared mightily, with a great well of Crown blood, a great stack of parchment, and an even greater stack of cheese rounds. He was assisted in his special purpose by the Shield Maiden Blyneeth, also known as "Blyneeth Quickhands," "Blyneeth the Key Master," and "Matching Set Blyneeth." Most would be distracted from their work by Blyneeth's charms. But this is not that kind of story. Her job this night was to keep Z'Jargo's flagon topped with Dew of the Mountain, for a long night it was surely to be.
Z'Jargo hung a sign on the Barrelry: SHOPPE CLOSED, using old-timey spelling because tourists just eat that stuff up. He returned to his hovel, tied back the sleeves of his tunic, and set to work.
By now, his vecktor testing was efficient and swift. Z'Jargo retrieved a slip of parchment and a quill dipped in Crown blood and carefully copied a vector listed in his journal onto the parchment, tongue protruding from the corner of his mouth in careful concentration. He placed the parchment and his ma'atrix amulet onto the shrine of Vandermonde and intoned the sacred words. The transmutation complete, Z'Jargo retrieved the parchment and held it to candlelight for inspection.
Imagine now Z'Jargo working into the night, scraps of parchment and cheese round crusts littering the floor, a flagon of Dew of the Mountain at table's end. Outside there is the quiet trill of Frost Crickets and the bubbling of the nearby beck. Inside the work continues.
Z'Jargo writes [ ack pling ] on a scrap of parchment.
Transmuted, the parchment reads [ frond NAK ]. Failure. Z'Jargo strikes another entry from his journal.
Again.
Z'Jargo writes [ bang bucky ] on a scrap of parchment.
Transmuted, the parchment reads [ plic grok ]. Failure. Z'Jargo strikes another entry from his journal.
Z'Jargo reads the next vecktor from his journal, enters it on parchment, transmutes it. It too, changes. Then another and another. [ hash DAK ] becomes [ weenie banana]. [ tick pop ] becomes [ virgula thrope]. [ BIRK blugle ] becomes [ pipe PIPE ]. [ enyay cat ] becomes [ wham awk ]. Finally, in a fit of frustration he slams his big fists on the altar.
Great Herders of the Nerfs! Z'Jargo exclaims. Is there no end to this curs'ed curse?
The noise wakes Blyneeth. She appears behind Z'Jargo, rubs his shoulders. Quandoque bonus dormitat Vandermondus, she whispers. Come to bed, my love.
I am not tired.
Then perhaps I can tempt you with some Skeevy pie?
Not now, woman, Z'Jargo grumbles. I have no time for mammals. The inviting lady, fatigued by all his brooding, slips away.
Resolute intonation and occasional cursing continue. The threatening dawn brings Z'Jargo to the last page in his journal. Only two entries remain. The penultimate is written [ talky bit ] and had been discovered in a cave under the Sea of Spirits. The final entry was found carved into a Mammoth tusk frozen in the frozen wasteland of Blizzard Freeze.
Let us end this charade, Z'Jargo sighs. He glances at Blyneeth now sleeping in the corner wrapped in a FireWolf pelt.
Now he places a scrap of parchment upon the alter upon which is written [ talky bit ] and offers the vecktor [ talky bit ] to the EigenGods. Thereupon, the EigenGods return [ TALKY BIT ].
Z'Jargo sat blinking in disbelief.
A trick of the candlelight? No! For in repeating the offering the same result is returned. He had found an eigenvecktor!
Z'Jargo had considered this moment many times in his waking dreams. There it was accompanied by fanfare, bright lights, a heavenly choir, the hearty huzzah of dead elders. But here, as the rising sun shrank the night into shadows, there was only silence save for the breathing of Blyneeth. Z'Jargo possessed a living secret, a glimpse behind the curtain. He had begged Na'ature for a look up her skirt, and Na'ature had finally obliged him. And now, alas, discovery of discoveries and no one to share it with! A muffled thunderclap burst off in the distance. Or was it maniacal laughter?
He moved to wake Blyneeth, but she would not be roused. He ran outside and looked to and fro, up river and down river, but lo the town was also asleep, as the reasonable would be at such hour. It fell, then, to the unreasonable. To Neuman the Fool, who happened to be happening by Z'Jargo's hovel at this very moment. Surrendering to this strange irony, Z'Jargo described his extraordinary discovery to the Town Fool, in a loud voice and with much excited gesturing. Neuman almost certainly understood none of it. But that mattered not, for Z'Jargo felt as light as a Ravehawk's feather. Having told his discovery to another he was now certain it had not been a dream. As if by speaking the story aloud he had proofed its verity.
And so it was, in that hour, and for many hours after that, Z'Jargo and the Fool Neuman were the only two souls in all of YonderEdge who knew an eigenvecktor had been found.
Or so Z'Jargo thought... (dun, dun, dunn)
INTERLUDE
Some things we skipped over in Parte Four described here so we don't have to do a flashback later and also to let that ominous closing musical cue sink in a bit for dramatic effect.
After speaking with, or rather at, Neuman, Z'Jargo returned to experimenting. He verified his vecktor was indeed an eigenvecktor. And again. And once more. The transmuted parchments he lay side-by-side only to notice that components of the transmuted vector appeared to exhibit a constant scaling of their corresponding original entries. Z'Jargo hurriedly retrieved his barreling calipers, for these would provide accurate measurement. He carefully measured pairs of runes, noting their size before and after transmutation. Having made a summary of the changes he quantified the change in size, closing one eye and holding his thumb out at arm's length. After a time of this he was convinced: each entry in an eigenvector was resized by the same amount. There was a common magnifying facktor!
A second important discovery! All in the space of a few hours, after half a lifetime of searching.
What more? What more? Z'Jargo's mind whirled, his thoughts a train in a Golan-Globus film. It was then he realized: there was a vecktor yet in his questing journal he had not tested!
On any other night, he would have hesitated. The final test at the end of a night is enough to drive even the most dedicated Algebraist to fatal procrastination. There are dishes to wash. A hound to walk. Perhaps it is time to re-thatch the roof, no? Surely, Vandermonde would not smile twicely in one night!
But Z'Jargo forced himself forward. He ran his thick finger down the page of the journal, past the penultimate entry -- his jewel -- and onto the last line of the page. This he entered on parchment and gave to Vandermonde.
Great Hessians and Jacobians! This, too, was an eigenvecktor! Z'Jargo checked it again, to make sure there was no mistake. But there was no mistake. A third discovery: a ma'atrix may have more than one eigenvecktor. He retrieved his calipers and measured the scaling facktor for this eigenvecktor. He was almost delirious from lack of sleep. His vision blurred, his hand shook. But it appeared that this eigenvecktor, too, obeyed scaling, albeit the scaling facktor differed from that of the first.
Now Z'Jargo dared go no further, Exhausted, he fell into the FireWolf pelts and spooned Blyneeth. Soon he was asleep. He knew that when he awoke, the world would be forever changed.
PARTE FIVE A Strange Denouement
Master Strang, Z'Jargo greeted. This is most unexpected.
But Strang was short on pleasantries and pushed his way inside. Word has reached me of your discovery, he said.
It has? But how? Suddenly, Z'Jargo understood. Neuman! he muttered, clenching his fist.
Neuman indeed, Strang said. Not so much a fool as a calculating fool. Do you think I would leave you to such important work without keeping tabs? But come, tell me the tale entire, for I am wont to hear it.
Here Z'Jargo told Strang the Elder all that had transpired. The first eigenvecktor. The scaling facktor. The second eigenvecktor. Its different scaling facktor. The details of which are available in the Interlude provided above so we need not insert a flashback here.
Strang the Elder listened intently. When Z'Jargo was done, the two men sat in silence for long minutes, Strang staring off into the distance as if watching a montage of events to come. Finally, Master Strang spoke.
This facktor of scaling. Have you invented a proper name for it?
Strang circled behind Z'Jargo, but Z'Jargo took no notice.
Yes I have, Z'Jargo answered. I call it an "eigenvalue."
Eigenvalue, hmm?, Strang said, slipping a hand beneath his robes. Alas, you should have called it a DIE-genvalue!
Upon which Strang drew a dagger and fell upon Z'Jargo. Their struggle was not really as epic as you would hope. For while Strang was treacherous and crufty, he was an academic. Dagger or no, his slight torso and little garter snake arms were no match for Z'Jargo's muscly bulk, built up from his many years of barreling.
Stop it, you weirdo! Z'Jargo commanded. He ripped the dagger from Strang's hand. What is the meaning of this?
And here it was that the Chief of the Algebraist Brotherhood began to weep. His eyes red, his face wet, he spoke in many sentences ending in an exclamation point.
You fool! You have undone us all! For once the secret of the eigenvecktor is known, there will be no more need of it! School children will learn its making. It will become known to wizards, to sorcerers, to.. to Art History majors! And then the Steam Elves will build a machine that extracts eigenvecktors from a ma'atrix using nothing more than steam and spinning gears and wheels and steam! The Brotherhood will be no more! We will be obsolete!
Z'Jargo calmed his Master. He sat him at the table, brought him a flagon of Dew and some snoutloaf.
Z'Jargo then recounted the tale of his boyhood. How his father was a Barreler, and his father before him. When his grandfather arrived, Z'Jargo explained, the villagers didn't have barrels. They just put their stuff on the ground and drew a circle around it in the dirt. His grandfather invented the concept of "putting one thing in another" and built the first barrel ever. His father improved upon the design with the invention of the lid. His father also invented the Barrel Maker's Song, and Z'Jargo gave a stirring rendition.
Year after year the family business thrived, Z'Jargo explained, and the village thrived with it. Tribute was collected by the Knurl which allowed them to have clean drink-water, and open a school for the children, and pay tower-men who kept watch at the wall. They also built the finest wenchkeep in all the land. The drink-water was kept in barrels. The school required barrels. The watchmen required barrels. Even the wenches required barrels. More came to live in the village each year, all of whom were in need of barreling. Once, the High HogBorn demanded the Knurl stop collecting tribute, arguing that it would allow the village to grow more quickly. But the High HogBorn were smoted with cudgels and their broken bodies given to the ground. The villagers may live in a place with dragons and talking trees, Z'Jargo explained, but even they're not gullible enough to believe a Laffer curve.
It was here that Master Strang glanced at his watch. You wanna try landing this thing, son?
My point, Z'Jargo concluded, is that the Brotherhood will never be obsolete. Z'Jargo held out his great gnarled hands, making the universal gesture for barrel. Knowledge is like a barrel. The more you put in them, the more barrels you need.
That's the dumbest analogy I've ever heard, Strang observed.
Hey, I didn't see you figuring out the whole eigenvalue thing.
Meh, fair point. The old Master pushed his empty plate aside and stood to take his leave.
And so here we will tell you a little of the future. Z'Jargo became a tenured member of the Algebraist Brotherhood. Soon, other discoveries built upon his fateful breakthrough. They found that a ma'atrix sometimes has more than one eigenvalue and/or more than one eigenvecktor and sometimes not. They found that many a ma'atrix has as many eigenvalues as the size of the ma'atrix. They found that by arranging the eigenvecktors themselves in a ma'atrix and applying this ma'atrix twicely to the original ma'atrix, it itself became transmuted to a diagonal ma'atrix. And many things besides. But those are stories for another time.
Z'Jargo slapped Master Strang heartily between the shoulder blades. Come, let us away to the wenchkeep.
Verily, they have an early-bird special.
EPILOGUE
In which we tease the reader with the possibility of a sequel
Klaxxon the Skittish worked third shift at DuskGuard, tasked with keeping vigil for putative Vampires skulking around the darker places of YonderEdge. He only took the job for the money, as he had recently married the maiden Athena and bought a house and was thinking of adopting an orphan or two. (One wonders why they can't obtain children in the usual fashion, but that is a story for another time.) Such quests demand substantial coin, and third shift paid well. So here he sat, shivering on the stony ramparts night after night, staring off at nothing in the lands beyond.
Vampires my Bu'utt! Klaxxon scoffed. What's next? Gargoyles coming to life? Horse ghosts? Bah!
But on this night there came a visitor of a different sort.
It came from the dark side of the moon, it seemed, and made the moon vanish. Not as if obscured by clouds, but vanish suddenly, as one might extinguish a torch. A great shadow carved from moonlight fell all along the watchtower.
Klaxxon looked to the sky and saw a figure blotting out the firmament. The figure of an enormous dragon!
Before Klaxxon could stand the beast was upon him. The ground shook and the trees swayed and stones fell from the tower wall and rolled down the steps like tumbling dice. The air became a choking swelter, Klaxxon's vision wavering in the crenellated heat. The crickets stopped their cricking and frogs their croaking. Now there was only the sound of a living dragon. It stood, it stamped, it flexed its terrible wings.
It looked right at poor Klaxxon!
At that moment Klaxxon believed he would perish. Baked, skewered, eaten. And whatever all else dragons did.
But the dragon did not eat Klaxxon. Instead, it spoke.
Where? Where is the Eigenborn?
The dragon's voice was like a shattered glass in an acid bath. Klaxxon was so frightened he could not move. But he summoned the courage to answer, for not answering would only insult the beast, and then a lot more than his thumb would be bitten.
The Who?, Klaxxon squeaked.
The Eigenborn! Where?
Klaxxon's face scrunched up. You mean Z'Jargo?
The dragon stamped and howed and reared back. It spit a great pillar of flame into the sky that flashed night to day and dispersed the clouds overhead as if they had been shouted away. The true name of the Eigenborn was like an arrow stabbing through a convenient flaw in its scaly scales!
Yes! it roared. The Eigenborn! Where?
He... he lives with Blyneeth at SnowFall College. Klaxxon extended a shaky arm and pointed toward the northern horizon. Her speech tree perks helped him negotiate a two-and-two with the department. Nice place. Top-notch Art History program--
Silence! the dragon commanded. With that, it leapt into the air and with a single beat of its wings turned north. For the location of the Eigenborn was all it wanted to know and now having been told the location of the Eigenborn it concluded its business with Klaxxon. This dragon was not the sort of Chatty Cathy dragon you find in tales nowadays, who need half a trilogy and a theme-park ride to get their point across.
Klaxxon called after the beast as it flapped away. Wait!, he called out. Who are you? What is your name? It was as if he had quaffed a Potion of Curiosity.
My name? the dragon roared, its gaze upon the far horizon as constant as the northern star.
I am Ennemond! it bellowed.
But the Eigenborn will know me as Jordan.
Image Credits
Images and screen captures from The Elder Scrolls series of video games and from websites created and owned by Bethesda Softworks, the copyright to which is held by Bethesda Softworks. Use of these images in a parody of the original work is claimed as "fair use" under applicable copyright law. Still, feel free to purchase a fine Bethesda product this very day, which might stay their legal team from going medieval on me.
Images may have been cropped, color corrected, and/or otherwise modified from their original form.
ESSAYS IN LINEAR ALGEBRA by Strang the Elder not available as an in-game item.
Images and screen captures from The Elder Scrolls series of video games and from websites created and owned by Bethesda Softworks, the copyright to which is held by Bethesda Softworks. Use of these images in a parody of the original work is claimed as "fair use" under applicable copyright law. Still, feel free to purchase a fine Bethesda product this very day, which might stay their legal team from going medieval on me.
Images may have been cropped, color corrected, and/or otherwise modified from their original form.
ESSAYS IN LINEAR ALGEBRA by Strang the Elder not available as an in-game item.
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