Chapter seven
I owe Chapter 2 a debt, and this is the chapter in which I pay it.
In Chapter 2, I described, briefly and a little dryly, the kind of adult I had constructed for myself. The construction was simple. I had observed my father. I had reached the only conclusion the data supported, which was that becoming him was a fate worth any amount of personal redesign to avoid. The redesign duly began. He drank; I became a teetotaler. He slept around; I made loyalty the load-bearing virtue of my life. He was lazy; I became ambitious in a way that, viewed from any angle, looked indistinguishable from being chased. He destroyed himself; I refused to destroy myself, which is not the same as being well, but which is, I will grant, a reasonable opening move.
I lived inside this construction for approximately three decades. I was, by every external measure, succeeding at being his opposite. I was also, by every internal measure, exhausted in a way I could not name, and the energy required to maintain the construction was, on inspection, almost exactly the energy required to feel exhausted all the time. I had assumed this exhaustion was the price of doing well. I had been wrong about the price. I had been wrong, more interestingly, about the doing well.
This chapter is about why. The why turns out to be a piece of linear algebra so simple that, once you see it, you cannot quite believe nobody told you about it earlier. The piece of linear algebra is called an eigenvector, and it is, on inspection, the cleanest mathematical description I know of what selfhood actually is.
I am going to introduce eigenvectors without any of the algebraic machinery that makes them difficult in undergraduate textbooks. There will be no determinants in this chapter. There will be no characteristic polynomials. There will be one picture, one observation about the picture, and a long quiet argument about what the observation means for the kind of person you have, for many years, been pretending to be.
Here is the picture.
Imagine a flat plane. On the plane, draw a small grid of arrows, each one pointing away from the origin in some direction. The arrows represent, for our purposes, the various impulses, instincts, preferences, and inclinations that, together, constitute a personality. Some arrows point toward solitude. Some arrows point toward company. Some arrows point toward risk. Some toward safety. Some toward the past. Some toward the future. Each arrow has a direction and a length. Each arrow is a small piece of who you are.
Now imagine that you are about to be acted on by something. Call this something a transformation. The transformation could be almost anything in real life. A parent's behavior. A traumatic event. A relationship. A culture. A decade of working in a particular kind of office. A friendship that lasts twenty years and slowly rewrites the shape of you. Whatever it is, it acts on every one of your arrows at once, and it changes them. Mathematically, a transformation is a rule for taking each arrow and producing a new arrow. Some arrows will be lengthened. Some will be shortened. And, crucially, almost all of them will be rotated. They will end up pointing in different directions than they started.
This is what most life events do to most parts of your personality. They rotate you. You used to want one thing. After the event, you want a slightly different thing, in a slightly different direction. You used to fear one thing. After the event, you fear it in a tilted way, mixed with other fears that were not present before. The transformation has acted. The arrows have moved.
Now here is the observation. It is the observation that the whole chapter rests on, and I want you to register it carefully.
In almost every transformation, no matter how violent, there are a small number of special directions in which the arrow does not rotate.
Read that again. The arrow may be stretched. The arrow may be shrunk. The arrow may, in some cases, even be flipped to point exactly the opposite way. But the arrow remains on the same line it started on. It still points in, or against, the same direction it originally pointed in. The transformation has acted on the arrow. The arrow has not changed direction.
These special directions, the ones the transformation cannot rotate, are called the eigenvectors of the transformation. The word eigen is German. It means own, in the sense of belonging to oneself. The eigenvectors of a transformation are the directions that, in some technical and ancient sense, belong to themselves under that transformation. The transformation can scale them. The transformation cannot turn them.
Figure 7.1 Before and after a transformation. Most vectors end up pointing in new directions. The eigenvectors, drawn in accent color, are stretched but remain on their original lines.
The figure makes the geometry visible. Notice the two accent-coloured arrows. They have, in the after panel, been stretched by the transformation. They have not, however, been rotated. They are still on the same lines they started on. They still point in the same directions. The transformation has acted on them, in the strict sense, but it has not turned them. It has lengthened or shortened them along their own axes, and that is the most it can do.
That is what it means, mathematically, to be an eigenvector of a transformation. To be the direction the transformation cannot rotate.
Now I am going to make a claim, and I want you to test it against your own life as I make it.
You, as a person, are walking around with a small number of eigenvectors of your own. They are the directions in your personality that life cannot rotate. Life can stretch them. Life can shrink them. Life can, on bad days, even flip them. But life cannot make them point somewhere they were not, in some original and ancient sense, already pointing.
Consider the things that are most stably true of you. The kind of humor you find funny when you are not trying to be impressive. The kind of work you find interesting when nobody is paying you to find it interesting. The kinds of people you are drawn to in a room. The thing you reach for, instinctively, when you are bored. The shape of your curiosity. The texture of your tenderness. The specific way you, and only you, lose patience.
These are the eigenvectors. These are the directions that, no matter what life has done to you, you keep coming back to. The arrows may be stretched or shrunk by what has happened. The arrows still point where they pointed.
The rest of you, the personality you wear in public, the way you sit in meetings, the topics you bring up at parties, the partner you chose, the career you ended up in, the politics you have come to hold, all of this is, in linear algebra terms, a rotated vector. It is the result of who you started as multiplied by the transformation life applied. It is real, in the sense that it is happening, but it is not invariant. Change the transformation, by which I mean live a substantially different life, and the rotated vector will rotate again. The eigenvector, meanwhile, will be where it has always been.
This distinction, between the parts of you that are eigenvectors and the parts of you that are merely rotated outputs, is, I want to suggest, the most useful single distinction the anxious mind can learn to make about itself. The anxious mind, by default, treats every part of itself as load-bearing. It cannot tell which directions are eigenvectors and which directions are just the current outputs of a transformation it never chose. It defends, with equal energy, the eigenvector preferences and the rotated outputs, and it experiences any threat to either as a threat to the self.
This is exhausting, and unnecessary, and once you can see the difference, the energy budget gets a great deal smaller.
I owe you the resolution of the not-father algorithm, which is, in linear algebra terms, the cleanest possible illustration of a person who has confused a rotated output for an eigenvector.
Here is the geometry of what I had done. My father existed in some direction in the space of possible men. I had, at four, and then at eight, and then at fourteen, observed his direction and concluded, on the basis of evidence I had no reason to dispute, that it was a direction I did not wish to occupy. I had then constructed an adult by, very carefully, pointing myself in the exact opposite direction. He drank; I refused alcohol entirely. He was unfaithful; I was monastically loyal. He was idle; I was a man who did not know how to put a working day down.
I had thought, for the longest time, that what I had done was choose my own direction. I had not. I had chosen the direction that was minus-one times his. In linear algebra terms, I had constructed a vector that was a scalar multiple of his vector. The scalar was negative one. The vector was, in every other respect, on the same line.
This is the geometric problem with defining yourself in opposition. You and the person you have defined yourself against are, mathematically, on the same axis. You point in opposite directions along that axis, but the axis is the axis they chose. The transformation that life applies to that axis will affect both of you, in equal and opposite ways, but neither of you is free of the axis itself. You are still living inside their coordinate system. They picked the orientation. You picked, in the most limited sense, the sign.
This is why the not-father algorithm produces exhaustion. The man running it is doing the work of being against, full time, on every available axis, and the axes are not even his. They were given to him in childhood by the very person he is trying to leave behind. He is, in the most literal mathematical sense, still in a relationship with his father. The opposite of love is, geometrically speaking, not hatred. The opposite of love is indifference, which is to say, an arrow that points in a different axis entirely, in a direction the original transformation had no opinion about.
The fix, when I finally found it, was not to add more anti-father into my personality. The fix was to start finding the directions in myself that my father had never lived along. The directions he had no opinion about. The directions that were, in linear algebra terms, in his nullspace, which is to say, the directions his particular transformation simply could not see. Those directions, when I found them, did not feel like rebellion. They did not feel like victory. They felt, embarrassingly and quietly, like me.
I started, late in my thirties, to do things that my father had no theory of. I read poetry, which he had no opinion about. I learned to cook food I had not grown up eating, which he would not have known how to disapprove of. I made friendships with people who did not in any sense remind me of anybody from my childhood, because nobody from my childhood resembled them. The friendships were not against anybody. They were not for anybody. They were on a new axis entirely.
This was, mathematically, the first time in three decades that I had located one of my own eigenvectors. The eigenvector was not loud. It did not look like anything in particular. It was simply a direction in my own personality that nobody had ever made me hold an opinion about. The direction, when I sat in it, felt like nothing at all. Which is, in retrospect, exactly what selfhood feels like when it is finally not in argument with anybody.
Here, in Python, is the technical version of what I just described. The function takes a person's set of directions, applies a transformation to them, and returns both the directions that rotated and the directions that did not.
import numpy as np
def find_eigen_directions(transformation):
"""
Given a 2D transformation expressed as a matrix,
find the directions that the transformation does not
rotate. These are the eigenvectors. Their scaling
factors are the eigenvalues.
Inputs:
transformation: a 2x2 numpy array
Returns:
a list of (eigenvalue, eigenvector) pairs.
"""
eigenvalues, eigenvectors = np.linalg.eig(transformation)
return list(zip(eigenvalues, eigenvectors.T))
# A transformation that stretches along the x-axis
# and squishes along the y-axis. Like a particular kind
# of difficult childhood.
T = np.array([[2.0, 0.0],
[0.0, 0.5]])
for value, vector in find_eigen_directions(T):
print(f"direction {vector.round(2)} is scaled by {value}")
# Output:
# direction [1. 0.] is scaled by 2.0
# direction [0. 1.] is scaled by 0.5
#
# Two eigen-directions survive. One is stretched. One is
# shrunk. Neither is rotated. Both still point where they
# always pointed. They are who you were, before and after.The code is not what matters here. What matters is the abstract fact the code computes. Every transformation, no matter what shape it takes, has a small number of directions that survive it without rotation. Your job, if you wish to be free in any meaningful sense, is to find which of your own directions those are. Not by argument. Not by force of will. By the slower and more patient process of noticing, in your actual life, which of your inclinations have stayed exactly where they were, no matter what the years have done to the rest of you.
These are your eigen-self. They are the parts of you that nothing has been able to turn. They are, almost by definition, the parts of you that are most worth trusting.
I want to close the chapter with a small honest observation, because the cheap version of this argument is wrong, and the cheap version is, again, the version that ends up on a poster.
The cheap version is, be your authentic self. The cheap version fails because the anxious mind does not, by default, know which parts of itself are authentic and which parts are rotated outputs of transformations it never chose. The instruction to be authentic is, in practice, the instruction to be more of whatever you currently are, which may or may not include any actual eigenvectors at all.
The real version is quieter. The real version says: start looking, with mathematical patience, for the directions in yourself that nothing has been able to rotate. They will be unobtrusive. They will not feel like victories. They will feel, mostly, like the things you do when nobody is watching, including, importantly, you.
The eigenvectors are not the loud parts of your personality. The loud parts of your personality are almost always rotated outputs, because loudness, in psychological systems as in physical ones, is the signature of energy being applied. The eigenvectors are the quiet, unspectacular preferences that have been with you since you were small, that no transformation has been able to make you stop having, and that you have, on some bad days, been ashamed of, because they do not match the personality you have been performing.
They are who you are. They have always been who you are. The work, for the rest of your life, is not to add more of them. The work is to stop drowning them out with the rotated outputs you have been mistaking for the self.
A small exercise
Find one of your eigenvectors.
Think back to a time before adolescence, when you were doing something purely because you wanted to. Not for a grade, not for an audience, not for anybody's approval. Something specific. A book you reread. A way of arranging things on a shelf. A walk you used to take alone. A small obsession nobody else shared.
Now ask, as honestly as you can, the following question. Is some version of that thing, in some form, still true of me today?
Almost certainly yes. Almost certainly the form has changed. The grown-up version no longer rereads the same book. The grown-up version arranges different shelves. The grown-up version walks different routes. But the underlying direction, the eigen-direction, is almost certainly still on the same line it was on at nine.
Write that direction down. Not the activity. The direction. The shape of the impulse. I like to do this thing alone. I like to come back to the same object. I like to put things in order. I like to be outside. Whatever it is.
You have just located one of your eigenvectors. There are more. They have been with you all along. They are not loud. They have not asked for your attention. They have, however, never once stopped being true of you. Trust them, very gently, more than you trust the rotated outputs that have been performing in their place.
Chapter 8 takes a different turn, into probability theory, and into the question of why a Bayesian prior installed in childhood is, mathematically, almost impossible to update later in life. It will close, finally and fully, the loop on incidents I and III from Chapter 2, on the school that decided I was not going anywhere, and on the long marriage in which I refused to have children because I had performed, at fourteen, a piece of statistical inference I had no way of knowing was wrong.
For now, the page closes here. The eigenvectors are the directions life cannot rotate. They are quieter than the rotated outputs. They have, all along, been who you actually are.