Matrix Algebra Rule 34

Context: I said this; Bob O’Hara said this.

And then I said to myself, “bloody hell, what do you mean there is no Rule 34 of matrix algebra? Is there really no lazy mathematics graduate student who would— oh, wait.”

* * *

There is no 34th rule in matrix algebra.

No, the rules of matrix algebra are not numbered. They are wild, free, potentially uncountably infinite. They are discrete bricks of conditional truth baked from the raw red sludge of matricular concepts; bricks, and sculpted marble columns and colonnades that, founded on logical truth, reach up at a potentially limitless sky, supporting churches and palaces of proposition and conjecture. They make a city fair, ancient and beautiful.

The ghost labourers of that city’s construction are mathematicians; their sweat and tears are the mortar, their puny mortal minds the timid flesh-architects, unseenly present there to build this spirit world — or to make it reveal itself, if that is your philosophical inclination. Maybe these ideas pre-exist; maybe they are created by discovery. Maybe the distinction is bogus and meaningless.

Nonetheless, this fair city of the truths and potential truths of matrix algebra is vast. It sways upwards in most un-cathedral-like fractal growth, results building on results, outlines filled in, and new outlines mapped as mirrored, distorted translucent copies of those that already exist.

If you are a mathematician and study matrix algebra, you can see this fair city growing, alive, unfurling like a flower, uncurling and growing like a child, its growth accelerated thousandfold by your encounter of it in a book that has the labour of centuries behind it. Tens of thousands of mathematicians, or some bounded from above number of a meaningless magnitude, have each made their contributions, some minuscule, of epsilonian size; some sprawling giant tangles of invention-discovery-organization, and out of all of them is curated and arranged the seemingly easy progression that is a book on matrix algebra, that guidebook to a city of lovely dreams — and this is one of the stories it tells.

The city is one of pink marble towers and golden steps, one of many cities on the trembling mathematical globe; and it is arranged in sweet spirals of repeating patterns, laws inexorably echoing in ever different cases, lemmata-chapels kneeling humbly beside towering theorem-cathedrals, and swarming crowds of matrices funnelled hither and thither, sure and confident in their knowledge of the rule of law, and a law of many rules.

But in all that structure, there is no numbering of rules; and thus there is no Rule 34 in matrix algebra.

No, that rule is one of life, not of mathematics. Matrix algebra tells you you can add together two matrices of equal size, or multiply them — the rule, of life, tells this too can be done with passion, excitement, and, dare we say, sexiness.

Let A be the matrix defined by

\displaystyle A = \left[ \begin{array}{cc} a_1 & a_2 \\ a_3 & a_4 \end{array}\right].

Observe this matrix, dear reader: a simple country matrix — square, decent, with no particular qualities along any moral axis; no secrets that any decomposition might reveal. Let us lift its skirts, and with chaste passion observe the most succulent number a_1a_4 - a_2a_3 to be not zero; we shall not want more detail from our blushing everymatrix. It means our matrix A is by no means singular; but though common, it is lovely.

But ah! Tragedy strikes. Our common, nonsingular matrix has an inverse: the Aristotelian other half of its Hamiltonian soul, a matrix A^{-1}, made at the other end of the universe of 2×2 matrices; all different, impossibly distant — but by a chance of statistics they meet, and as is known, opposites attract.

Their romance kindles like every phase of the Gauss-Jordan reduction of a 1000×1000 matrix going off at once!

This is their downfall. Their love for each other is fierce, undeniable. But A is a Capulet; A^{-1} a Montague. They should never meet; all Verona of matrix algebra knows what their embrace will bring.

At first, their romantic play seems harmless, though intoxicating. They swap sweet nothings, hold hands (metaphorically speaking), go side by side. The sum A + A^{-1} forms, and sits inertly, sweetly, unsimplified.

They share a first kiss; the sum dissolves into passionate summation, a heart-pounding, bracket-clutching, element-interleaving rush of the first base, and the second. The sum is resolved — for a fleeting moment there is no A, no A^{-1}, but merely this sweet sight:

\displaystyle \left[\begin{array}{cc} a_1+a_1^{-1} & a_2+a_2^{-1} \\ a_3+a_3^{-1} & a_4+a_4^{-1} \end{array}\right] .

But alas, this happiness is not to last. They are interrupted! A foul cretin, a singular old matrix of evil aspect, sees the two, and runs to inform, to speculate, to conjecture, with no decency or peer review, on other, more unseemly operations the two might have engaged in. It has no shame — no mercy — it sees nothing but the trivial thrill of a basic operation in the actions of two young and innocent matrices in love.

The two push apart, alarmed, their determinant-crossed fate clear to them. There they stand,

\displaystyle A = \left[\begin{array}{cc} a_1 & a_2+\epsilon \\ a_3 & a_4 \end{array}\right]

and

\displaystyle A^{-1} = \left[\begin{array}{cc} a_1^{-1}&a_2^{-1}-\epsilon\\ a_3^{-1}& a_4^{-1}\end{array}\right],

a little epsilon of lipstick transferred from one to the other. Their brackets heave; their elements shudder, torn between a moment’s lust and happiness, and the sure knowledge of impending doom.

“We are done!” A exclaims. “Finished! Run with me, away, away from this pestilential Verona of matrix algebra, this place that will not tolerate our love! If they shall not have us, we shall not have them (I can prove this) — come! I care not if you choose complex analysis, or potential theory, or some cold and distant L^p space, where dim Hilbertian stars wheel overhead. If you but be with me, my dimensions shall remain unchanged, my elements fixed in their positions — come with me.”

“Oh, darling!” the grief-struck A^{-1} cries, and brushes the epsilon of its lipstick from A’s cheek. “I cannot. I must not. I would perish — you would perish — there are limitations to what the gods of Thales and Bourbaki allow us. Would a matrix fit in at the courts of the Functional King? No, we would be pleasure slaves, freakish exotics, not ourselves; we would perish. Would even the brotherly dukes of Geometry shelter us? Us, who are cold and hobbled imitations of them, to them — as they to us are a cipher of lines, circles, sizeless points — there is no life for a matrix, or two of them, save in matrix algebra.”

“Say not so!” A cries, his brackets near bursting with agony. “Do not require of me the mockery of others — no, require that, but do not ask me to live separate from you. We are inseparable matrices. (Poetic license.) I would sooner die.”

“Then let us both do so!”, A^{-1} cries, wild, fey: “Let us die together, our love consummated in a fatal embrace. Let us perish; for to perish together is sweeter than any lonely existence could be!”

And so they rush at each other, rows and columns tangling, multiplying in a wild frenzy of passion free of all care. A gasp escapes A as its virginal columns part; A^{-1} murmurs a comfort, and slides its rows alongside, inside.

“Oh!” A gasps; A^{-1} is a perfect fit. They are made for each other. Each row clutches a column, and each column rubs against a trembling virginal row; a_1 rubs alongside a muscular column, till it finds the hungry touch of a_1^{-1}; now A^{-1} gasps, too, frightened, eager, demanding and adoring, and surrenders to the pleasures of calculation.

Their multiplication is hurried by necessity, but there are no errors in it, Their elements dance, their brackets merge; the movement achieves an breathless symmetry, and the two coalesce into a single form, melting, swirling, snapping together, each part finding its corresponding part in the other, their worries melting into happiness, feverish happiness, impossible happiness, perfect happiness; their elements are but the very basic numbers of the mathematical universe, engaged in the most ancient actions of all mathematical life: the numbers, they multiply, they are added; the orgasmic pleasure of unification is beyond words — and it is deadly.

With a cry of unspeakable contentment, the two are gone, their love the doom and fate of them. It was as the cold stars of logic had decreed —

\displaystyle AA^{-1} = \left[\begin{array}{cc} a_1 & a_2\\ a_3&a_4\end{array}\right] \left[\begin{array}{cc} a_1^{-1}&a_2^{-1}\\a_3^{-1}& a_4^{-1}\end{array}\right] = I,

nothing more, nothing of them remains, but this: a single I, an empty shapeless thing, the identity matrix, a dead thing, an epitaph, a cenotaph:

\displaystyle I = \left[ \begin{array}{cc} 1&0\\ 0&1\end{array}\right]

— no sign of the lovers remains in it, but bare entropiac blankness; so it is not by that stone, but by this tale, that you, dear reader, are to know there were two young passionate matrices that once loved, and dared, and lost, in the old Verona of matrix algebra.

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