Linear Transformations will take you on a Trip Comparable to that of Magical Mushroom Sauce, And Perhaps cause More Lasting Damage
Long after I was supposed to “get it”, I finally came to understand matrices by looking at the above pictures. Staring and contemplating. I would come back to them week after week. This one is a stretch; this one is a shear; this one is a rotation. What’s the big F?
The thing is that mathematicians think about transforming an entire space at once. Any particular instance or experience must be of a point, but in order to conceive and prove statements about all varieties and possibilities, mathematicians think about “mappings of the entire possible space of objects”. (This is true in group theory as much as in linear algebra.)
So the change felt by individual ink-spots going from the original-F to the F-image would be the experience of an actual orbit in a dynamical system, of an actual feather blown by a bit of wind, an actual bullet passing through an actual heart, an actual droplet in the Mbezi River pulsing forward with the flow of time. But mathematicians consider the totality of possibilities all at once. That’s what “transforming the space” means.
What do the slots in the matrix mean? Combing from left to right across the rows of numbers often means “from”. Going from top to bottom along the columns often means “to”. This is true in Markov transition matrices for example, and those combing motions correspond with basic matrix multiplication.
So there’s a hint of “causation” to this matrix business. Rows are the “causes” and columns are the “effects”. Second row, fifth column is the causal contribution of input B to the resulting output E and so on. But that’s not 100% correct, it’s just a whiff of a hint of a suggestion of a truth.
The “domain and image” viewpoint in the pictures above (which come from Flanigan & Kazdan about halfway through) is a truer expression of the matrix concept.
Matrices aren’t* just 2-D blocks of numbers — that’s a 2-array. Matrices are linear transformations. Because “matrix” comes with rules about how the numbers combine (inner product, outer product), a matrix is a verb whereas a 2-array, which can hold any kind of data with any or no rules attached to it, is a noun.
* (NB: Computer languages like R, Java, and SAGE/Python have their own definitions. They usually treat vector == list && matrix == 2-array.)
Linear transformations in 1-D are incredibly restricted. They’re just proportional relationships, like “Buy 1 more carton of eggs and it will cost an extra $2.17. Buy 2 more cartons of eggs and it will cost an extra $4.34. Buy 3 more cartons of eggs and it will cost an extra $6.51….” Bo-ring.
In scary mathematical runes one writes:
And the property of linearity itself is written:
Or say: rescaling or adding first, it doesn’t matter which order.
“ADDING” “THINGS”
The matrix revolution does so much generalisation of this simple concept it’s hard to imagine you’re still talking about the same thing. First of all, the insight that mathematically abstract vectors, including vectors of generalised numbers, can represent just about anything. Anything that can be “added” together.
And I put the word “added” in quotes because, as long as you define an operation that obeys commutativity, associativity, and distributes over multiplication-by-a-scalar, you get to call it “addition”! See the mathematical definition of ring.
The blues scale has a different notion of “addition” than the diatonic scale.
Something different happens when you add a spiteful remark to a pleased emotional state than when you add it to an angry emotional state.
Modular and noncommutative things can be “added”. Clock time, food recipes, chemicals in a reaction, and all kinds of freaky mathematical fauna fall under these categories.
So “adding” is perhaps too specific a word–all we mean is “a two-place input, one-place output satisfying X, Y, Z”, where X,Y,Z are the properties from your elementary school textbook like identity, associativity, commutativity.
So your imagination is usually the limiting reagent in defining “addition”.
But that’s just vectors. Matrices also add dimensionality. Linear transformations can be from and to any number of dimensions:
1→7
4→3
1671 → 5
18 → 188
and X→1 is a special case, the functional. Functionals comprise performance metrics, size measurements, your final grade in a class, statistical moments (kurtosis, skew, variance, mean) and other statistical metrics (Value-at-Risk, median), divergence (not gradient nor curl), risk metrics, the temperature at any point in the room, EBITDA, notfunction(x) { c( count(x), mean(x), median(x) ) }, and … I’ll do another article on functionals.
In contemplating these maps from dimensionality to dimensionality, it’s a blessing that the underlying equation is so simple as linear (proportional). When thinking about information leakage, multi-parameter cause & effect, sources & sinks in a many-equation dynamical system, images and preimages and dual spaces; when the objects being linearly transformed are systems of partial differential equations, — being able to reduce the issue to mere multi-proportionalities is what makes the problems tractable at all.
So that’s why so much painstaking care is taken in abstract linear algebra to be absolutely precise — so that the applications which rely on compositions or repetitions or atlases or inversions of linear mappings will definitely go through.
Why would anyone care to learn matrices?
Understanding of matrices is the key difference between those who “get” higher maths and those who don’t. I’ve seen many grad students and professors reading up on linear algebra because they need it to understand some deep papers in their field.
Linear transformations can be stitched together to create manifolds.
You can’t understand slack vectors or optimisation without matrices.
JPEG, discrete wavelet transform, and video compression rely on linear algebra.
A 2-matrix characterises graphs or flows on graphs. So that’s Facebook friends, water networks, internet traffic, ecosystems, Ising magnetism, Wassily Leontief’s vision of the economy, herd behaviour, network-effects in sales (“going viral”), and much, much more that you can understand — after you get over the matrix bar.
The expectation operator of statistics (“average”) is linear.
Dropping a variable from your statistical analysis is linear. Mathematicians call it “projection onto a lower-dimensional space” (second-to-last example at top).
Taking-the-derivative is linear. (The differential, a linear approximation of a could-be-nonlinear function, is the noun that results from doing the take-the-derivative verb.)
The composition of two linear functions is linear. The sum of two linear functions is linear. From these it follows that long differential equations–consisting of chains of “zoom-in-to-infinity” (via “take-the-derivative”) and “do-a-proportional-transformation-there” then “zoom-back-out” … long, long chains of this, can amount in total to no more than a linear transformation.
If you line up several linear transformations with the proper homes and targets, you can make hard problems easy and impossible problems tractable. The more “advanced-mathematics” the space you’re considering, the more things become linear transformations.
That’s why linear operators are used in both quantum mechanical theory and practical things like building helicopters.
You can understand dynamical systems, attractors, and thereby understand love better through matrices.
The depiction of BDSM in popular films suffered a blow from which it will not easily recover with the release of Fifty Shades of Gray. While it was unfortunately many people’s introduction to the topic, bloggers from all corners of the internet have derided the relationship pictured in Fifty Shades for what it really is: abuse masquerading as kink. But twenty-four years ago, a family comedy centered on a couple who liked to torture each other for pleasure gave audiences a much healthier glimpse at BDSM.
Netflix describes the movie as “Stepping out of the pages of Charles Addams’ cartoons and the 1960s television series, members of the beloved, macabre family take it to the big screen.” Some scenes from the 1991 film The Addams Family are indeed straight out of the Charles Addams comic on which it’s based, like when the family douses a group of Christmas carolers with a cauldron full of steaming liquid. Others — like Morticia trimming the heads off of roses to arrange the stems in a vase — are exact recreations of the ‘60s TV series.
But what separates the film from the Family’s earlier iterations (besides, you know, colour) is the reciprocal nature of Gomez and Morticia’s relationship. The tired and offensive trope of an uninterested woman pursued by a lascivious man has appeared over and over again since the advent of television, and though Gomez and Morticia always exhibited a love and respect for each other stronger than nearly all TV couples, even the ‘60s version of Morticia had to rein Gomez in from time to time. Obviously this has a lot to do with the media mores of the time… but unfortunately, those sentiments still prevail today. And that’s why the The Addams Family film is so unique in its depiction of relationships.
The Addams’ lawyer Tully and his wife Margaret exemplify a sadly more familiar and cynical marriage: two people who ostensibly can’t stand each other but feel forced to stay together. The loathing is definitely mutual: when Margaret asks rhetorically, “Why did I marry you?” Tully responds, “Because I said yes!” The “unhappily married” cliché exists to varying degrees in most American media, to the point where Gomez and Morticia’s contrasting relationship is noteworthy.
The Addams constantly become enrapt with each other, getting sidetracked by each other’s allure, recalling their first meeting fondly, waltzing presumably numerous times a day. Morticia’s first lines of the movie, as the ever-present ghostly light with seemingly no source illuminates her eyes, describe Gomez’s sexual behaviour the night before: “Last night you were unhinged. You were like some desperate howling demon. You frightened me.” The camera zooms closer while she adds: “Do it again.” That’s right: the very first lines between the couple aren’t just a rare example of a man and woman who have been married for some time who can actually stand to be around each other. These lines, and the couple themselves, are an example of consensual BDSM.
The passion between the two has been famous since the television show, and the movie does an excellent job highlighting it as well. But unlike the ‘60s television show, Morticia seems as willing as Gomez to derail the conversation and submit to whatever distracting passion arises. The famous “Tish, that’s French!” lines are not, in the film, an example of Gomez’s passionate obsession with Morticia while she sighs and shakes her head happily. Morticia is an active participant and instigator when it comes to their conversation-stopping carrying-on. She’s just as happy as he is to make the others, and the audience, wait for the action to move forward, while they engage in behaviour more suited in media to new, young love than to a mom and dad.
Morticia takes it upon herself to confront Fester and initiate the film’s climax. The villains overpower her instead of listening to her, and strap her to a rack to torture her so that she’ll tell where the Addams family vault is hidden. Of course, following in the Too Kinky to Torture trope, Morticia isn’t phased by the stretching (she famously referred to the torture room as “the playroom” in the ‘60s TV series). Fester, however, is extremely anxious about hurting Morticia. The whole reason she allowed herself to be put in this “predicament” that for her is regular foreplay is so that Fester’s resolve would be weakened even more so against his overbearing and abusive mother. When Gomez turns up to “rescue” her, it’s less that she needed rescuing and more that Gomez needed the thrill and motivation to get out of his Sally-Jessy-Raphael-watching funk and defend his home. In this way, Gomez is more of the damsel in distress than Morticia ever will be. This is also the only time that Morticia dissuades her husband from continuing their flirting. As Gomez is loosening her straps while Fester confronts his mother, he’s clearly distracted:
Instead of scolding him like a typical wife character, Morticia reassures him that there will be time to continue the torture scene.
But what’s even more exciting, for me at least, is when Gomez and Morticia’s mutual attraction and respect is again evident in their kinky sex life. “Don’t torture yourself, Gomez,” Morticia orders: “That’s my job.” This movie doesn’t only offer an example of a loving, respectful BDSM couple — something painfully rare whenever kink is broached in film — but a loving, respectful, switch BDSM couple. That is to say, it seems as though each member of their exquisitely enviable partnership takes turns acting as the dominant and submissive role.
Much has been written in the blogosphere about what a good feminist role model Morticia is, and I agree entirely. But I would like to enthusiastically add that she takes the role of Dominatrix at least some of the time, and that it’s not played for a laugh or to emasculate Gomez. The passion, love, and respect the Addams couple famously has for each other extends to their role-reversing kinky sex life.
More than two decades later, filmmakers could really benefit from taking a page out of The Book of Addams and show us kinky couples who are also consensual, loving, and respectful. Though of course, none will approach the wonder that is her “mon sauvage” and his “cara mia.”
Recent Comments
Lillie W.: Ah Caroline, you understand these things so well… [Link]
I’ve been in love with the Addams family since I was a kid, and the amount of perfect this is, making new points to an old favourite, makes me want to cry. I love it.
Especially of note is the observation that Morticia and Gomez are switches, because I think there’s a general misunderstanding that this is not a thing.
Always Reblog Addamses. And this is but one of the many reasons why.
Jacob’s Ladder is the colloquial name for a bridge between the Earth and Heaven that the biblical Patriarch Jacob dreams about during his flight from his brother Esau, as described in the Book of Genesis. The story of Jacob’s Ladder is actually an ancient allegorical biblical tale describing the Alchemical process of reaching complete Gnosis or what some may call, Sainthood or Enlightenment. A Symbolic Ladder that we all must climb if we wish to reach the Spiritual Heights of the Divine in the Heavens while we are encased in Physical Matter here on Earth. As we climb, we must purify ourselves, our thoughts, habits and actions so that we may reach that seventh and final step of our ascent in order to activate all of our seven senses and DNA.
Computer graphics paper from Microsoft Research Asia demonstrates method to edit the light source of a photograph using neural network analysis:
We present a neural network regression method for relighting realworld
scenes from a small number of images. The relighting in
this work is formulated as the product of the scene’s light transport
matrix and new lighting vectors, with the light transport matrix
reconstructed from the input images. Based on the observation that there
should exist non-linear local coherence in the light transport matrix,
our method approximates matrix segments using neural networks that model
light transport as a non-linear function of light source position and
pixel coordinates. Central to this approach is a proposed neural network
design which incorporates various elements that facilitate modeling of
light transport from a small image set. In contrast to most image based
relighting techniques, this regression-based approach allows input
images to be captured under arbitrary illumination conditions, including
light sources moved freely by hand. We validate our method with light
transport data of real scenes containing complex lighting effects, and
demonstrate that fewer input images are required in comparison to
related techniques.
