OpenAI Is Joining Amazon in a nascent group of major tech companies to invest astronomical sums of money in machine learning projects that can best be surmised as “Tone Deaf.”

This example output from OpenAI’s new Jukebox demo probably illustrates it best:

OpenAI situates this track as the output of their waveform generating neural network, which has been conditioned on music by Kanye West.

The result is actually a very garbled hip hop audio with clearly audible excerpts from Eminem’s Lose Yourself. It is more revealing of Deep Neural Networks’ tendency to obscure and appropriate the labor used in their creation than it is impressive that they made Eminem sound a little bit like Kanye West.

As was the case for Amazon’s DeepComposer, George Lewis’ writing about music tech is probably the best lens through which to study OpenAI’s Jukebox. Lewis published a paper in the Leonardo music journal in 2000, which includes this quote:

“As notions about the nature and function of music become embedded in the structure of software-based musical systems and compositions, interactions with these systems tend to reveal characteristics of the community of thought and culture that produced them.”

We should understand the meticulously researched, astronomically expensive, machine learning technology on display here as representative of the “community of thought” at OpenAI.

I’ve been getting a lot of questions about collaborative online music in the age of Covid-19. There are lots of performance and DJ streams on Twitch and Youtube. What are the options for collaborating on making music. What is the lowest-latency way to play music together over the internet? Can my choir rehearse over zoom? How can I collaborate on recordings with other musicians while we are self-isolating? What follows is a quick brain dump that I hope will be helpful.

NOTE: The tweet-sized description of my PhD work at the MIT Media lab is: “Understanding the long-term potential of the internet, AI, and streaming technologies for music through experimentation” and I have way more to say about this than almost anyone wants to hear. I’ll try to keep this pragmatic.

No one wants to hear the short answer, but: Real-time (synchronous) internet-enabled music collaboration will never seriously compete with playing music with other people in the same room. Video conferencing options like Zoom and Google Hangouts have a combination of latency and quality issues, and will be more frustrating than rewarding. When making music together online, I recommend non-real-time (asynchronous or semi-synchronous) approaches.

Below I list a variety of collaborative technologies for creating music on the internet. I’ve written about this in the past, but FWIW, the outlined below technologies are primitive, and are (in their own different ways) attempting to squeeze historical ideas and formats into the online medium. We haven’t yet figured a way to create music collaboratively on the internet where the advantages outweigh the disadvantages.

However because many of us are self isolating right now, playing music in the same room just isn’t possible…. So now is a good time to look at some of the current options for online music collaboration.

### Realtime Internet Enabled Collaboration

If you really want to shoehorn the experience of making music together in real-time into the internet, your foremost option is JackTrip (Mac, Linux), (Windows). JackTrip is open source, free, and probably the best option for creating reasonably low-latency audio connections over the internet. It also scales to multiple concurrent users, as bandwidth allows.

I would go with JackTrip if everyone in your ensemble knows what an IP address is, and is ready to set up port-forwarding if needed. JackTrip requires that one user run a server, which will be easier if that user is familiar with a command line, Jack Audio, and has some knowledge of computer networking. JackTrip comes from Stanford’s Center for Computer Research in Music and Acoustics, which published a write-up of one of their early experiments, containing some useful references.

Note that all real-time options will have latency. Your mileage will vary. If your ensemble is small, everyone is in the same zip code, and everyone has a fast internet connection you might just get usable results.

A quick note about latency: Assume our latency threshold is 20 milliseconds. Anything higher than this and your ability to play together begins to suffer. Light travels approximately 3728 miles in 20 milliseconds (in a vacuum). Absolute best case scenario for round-trip audio is 1864 miles or NYC to Wyoming (but not across the Atlantic). Practical results will rarely be better than half that, and typically be much, much worse, due to network congestion, bandwidth limitations, IP throttling, the fact that light travels slower in the optical cables and internet hardware that it does in a vacuum, and other factors.

Experiments show that musicians can perform together acceptably with latency as high as 50 milliseconds. However, even in the unusual situation when you have a low-ish latency connection to your fellow musicians, there are other difficulties associated with playing music together over the internet in real-time. Think of how difficult it is just to get a Zoom meeting in the office going, with projectors, calls, constantly misbehaving in creative new ways. Now imagine that every participant brings with them the complexity of a small home studio. Suddenly trying to coordinate a performance becomes much harder.

### Almost realtime (with optional Zoom video share)

If your use-case can tolerate approximately 100 milliseconds of latency, and you want lossless audio, the subscription service by Audio Movers might be what you need. Their service uses an audio workstation plugin, and can stream to a sharable browser URL. The video below shows how to synchronize a web stream to a Zoom call for Audio+Video.

This might be useful in professional situations as discussed in the video. However it will probably not be suitable for realtime jamming or distributed performance.

### Non-Internet Network Audio

If you are lucky enough to be on the same physical network as your musical collaborators then JackTrip might work well. However, depending on your network infrastructure, you can probably get even better results with professional audio-over-ethernet options such as AVB (Supported by modern Motu interfaces like the UltraLightAVB) or Dante (supported by modern Focusrite audio interfaces like the REDNET2XP). These protocols can deliver high channel counts at sub-millisecond latency over a local network. For Dante or AVB, ensure compatibility with your network hardware before buying audio interfaces.

A number of startups are developing tools that add cloud enabled collaboration to the form factor of conventional digital audio workstations. These add real-time remote collaboration to a digital audio workstation, meaning that users record and edit audio at the same time similar to how users can edit text collaboratively in Google Docs. Note that the real-time interaction is for composing, editing, or recording music – these are not helpful if you want to play music together in real-time.

All of these are less mature and less capable than native digital audio workstations. Soundtrap was purchased by Spotify in 2017, and so it has a little more support than some of the others. It is also browser based, which means its sessions are easy to share, but limits its capabilities in other ways. These tend to be aimed at podcast creators, and beat makers, but you can use them for other kinds of music as well.

### “Dropbox for Digital Audio Workstations”

If you are accustomed to working with a conventional digital audio workstation like ProTools, Ableton Live, or Reaper, one easy option is to save your session files along with the accompanying audio files in a Dropbox folder, and then pass them off to your band mates so that they can record over them. Unlike the options above, only one user will be able to record or edit at a time.

Dropbox works okay for this purpose, but I would also recommend taking a look at these variants, which are specially designed for working with digital audio sessions. These include extra features like dedicated audio session version control and track preview. These can be helpful for sharing audio sessions between users.

These will try to sell you sample packs and drum loops. If you want to make music with samples or loops, this might even be useful.

### Mobile apps

These come and go so quickly it is hard to keep up! Here are just a few.

There are a million different mobile apps for creating videos. Want to make a video like the one that recently went viral by the Chino Valley Unified School District in California? Google video collage, and there are numerous apps that will help.

Acapella is one iOS app that cam make video collages, and includes built-in support for remote collaboration. Here’s a self-isolation themed music video made with Acapella.

View this post on Instagram

A post shared by AcapellaApp by Mixcord (@acapellaapp) on

TikTok is often used with video editor apps (like Acapella), but it also includes features for asynchronous collaboration called “duets” and “reactions” which are pretty self explanatory. If you know of a nice example of a musical TikTok Duet, please send me a link, and I’ll embed it.

Smule’s eponymous mobile app is specifically made to create karaoke style videos collaboratively. It includes a RockBand style vocal pitch display, supports remote collaborations, and can create video collages.

### Semi Real-Time Loop Based Jamming

This final category of apps is for jamming with people in almost-real time. Instead of trying to minimize latency, these increase latency to a musical duration in measures. They work kind of like playing with a looper, except that instead of layering over yourself, you are layering over what other musicians played a few bars ago.

• Ninjam is free and open source. It is designed to be used with the commercial, but very affordable DAW, Reaper
• Jammr is a commercial version of the same concept. Might be slightly easier to setup than Ninjam, but for most purposes, I recommend Ninjam

If you are familiar with (or interested in) algorithmic music or code-based composition, there are also tools for “jamming” semi real-time with code. The TidalCycles software package supports several different collaborative paradigms including shared editors for a google-docs like experience capable of generating MIDI messages with code (thanks to Manaswi Mishra for the tip).

### Latency Tolerant Music

Music and compositions that do not depend on precise shared rhythm can be tolerant to latent or distributed interaction. Playing this kind of music is one way to circumvent some (but not all) of the obstacles introduced by video conferencing. I participated in Pauline Oliveros’ “World Wide Tuning Meditation” via zoom, and while I wasn’t expecting to enjoy it, I was surprised. A review in the New York Times explains how the piece works.

For technology centric people (such as myself) it’s worth recognizing that what makes this kind of music work is not the technology: it is the community leadership of the organizers.

## Final Thoughts

There’s lots of historical examples, and a million apps I left out. If you think of an important category or example that I missed, please feel free to reach out and I’ll add it to the list. If you don’t know me personally, try @CharlesHolbrow (or email, etc.)

COVID-19 has and will continue to be challenging for musicians. Performing musicians (who don’t typically have extra money laying around) suddenly had their tours cancelled and their income eliminated. Venues are closing. If you are lucky enough to be self-isolating with loved ones or family members, I recommend making music with them. Also please reach out to friends and family who may be more isolated.

It can be tricky to understand how Kubernetes Services and Ingresses interact. The most important distinction relates to the concept of a Kubernetes Service.

• A Kubernetes LoadBalancer is a type of Service.
• A Kubernetes Ingress is not a type of Service. It is a collection of rules. An Ingress Controller in your cluster watches for Ingress resources, and attempts to update the server side configuration according to the rules specified in the Ingress.

For both Ingress and LoadBalancer resources, different Kubernetes providers (such as GKE, Amazon EKS, or bare metal) support different features. One of the things that makes Ingresses and LoadBalancers tricky is that your YAML manifest files might not be portable between different platforms and controllers.

Let’s talk about Services. One thing that clarified services for me is understanding how the different services build on each other. For example the ClusterIP is a simplest type. NodePort does everything that ClusterIP does (and more). LoadBalancer is another layer of capability on top of NodePort.

So the mental process when I need a Service is:

1. Am I trying to help my pods talk to each other? If yes, ClusterIP is enough. If not…
2. Am I trying to make my Service accessible on the public web (on a port above 30000)? If yes, NodePorts is enough (this is unusual). If not…
3. Am I trying to manage most public traffic coming into the cluster? If yes, choose an Ingress or a LoadBalancer. This is where things get tricky, because your options depend on the controllers that are available on your cluster.
• Load Balancers tend to be a little simpler than Ingresses.
• Ingresses might come with nice features like TLS/HTTPS termination and limited HTTP routing.

In my cluster I use the NGINX Ingress Controller for routing incoming HTTP requests to different services based on their Host HTTP header and url.

Make sure you understand what ingress controller is installed on your cluster (if any) and know that the YAML manifests for it are likely not portable to other Ingress Controllers — the YAML manifest that you give to the NGINX Ingress Controller might need to be pretty different than the manifest that you give to your GKE Ingress.

It used to be that when I started a new RPG I got this amazing feeling of excitement when I began to explore the world. The thrilling feeling of entering a new area and wondering what secrets could be hidden, waiting to be discovered.

Getting older, I experienced this less and less. The novelty of exploring a virtual world seemed to wear off as I got older. Then in 2011 I (along with the rest of the world) found Minecraft. The experience was fresh and exciting. A great implementation of a different approach to generating a virtual world. Something about knowing that world is infinite and unlike the world that anyone else is playing made exploration feel meaningful again.

Once I began to internalize the patterns of terrain in Minecraft the feeling of excitement began to ebb. It was still exciting to find diamonds, but the random terrain began to feel repetitive and similar. I wanted to find something not random in the world - I wanted to find something that felt intelligent. What if there were a way to stumble on the ruins of other users civilizations, decayed after many years of abandonment?

Could a hybrid approach to world building be the next step in the evolution of wanderlust in games? I started writing this game for mobile devices, but my skills at the time were not enough to implement the networking required to simulate the dynamic world I imagined. Then I found Meteor, a javascript framework that makes it very easy to synchronize client data. I re-implemented the game, migrating from the the Moai game engine to Meteor with the front-end entirely written in a custom canvas based game engine. Here is my lightning talk at a 2013 meteor meetup in San Francisco.

There were some pretty awesome features in the last meteor version. Users could write their own custom servers, characters could move between servers with a complex three way authentication handshake. But I wasn’t following Ken Levine’s excellent game development advice: “Stop trying to be smart, and start trying not to be dumb”. The engine and features were too complex, and it didn’t make sense for me to be maintaining my own javascript game engine. So in early 2017 I started re-writing the game again. This time the server is written in Go, and the client built on top of PIXI.js canvas engine. Both decisions have already paid off.

In one way or another I have been working on this game since the first mobile version 2012. It’s nearly always in some part of my mind – even when I am too busy to be actively developing. Whenever I have a spare moment my mind wanders back to the most recent development challenge. There’s some inexplicable force that drives me to keep at it. I envision a dynamic and evolving world in the cloud. A dwarf fortress for the next generation. A Minecraft for the future. A limitless RPG for that can fill others with the same wanderlust that I found in RPGs of my childhood.

### Early American Experimental Music

American composer Charles Ives (1874-1954) wrote many experimental pieces where he would selectively disregard certain 'rules' of music theory, and explore the sonic results. In an essay on his experimental music, Ives asked this question:

If you can play a tune in one key, why can't a feller, if he feels like, play one in two keys?

Ives' early polytonal experiments include pieces with a melody in one musical key and accompaniment in another key, and pieces with separate voices, each in separate keys. These experimental pieces were not published or performed until long after he wrote them, but Ives' ideas did appear in his less systematic compositions. More importantly his ideas influenced a generation of younger composers.

Among these younger composers was Henry Cowell (1897-1965), who in 1930 published New Musical Resources, a collection of largely unexplored musical ideas. The book extends the notion of multiple simultaneous musical keys polytonal music to multiple simultaneous tempos, polytempic music.

Can we make music with multiple simultaneous tempos? Certainly. But Cowell took the idea a step further. Assume we have multiple simultaneous tempos and these tempos are accelerating or decelerating relative to each other. In New Musical Resources, Cowell called this practice sliding tempo, and he noted that this would not be easy to do. In 1930, he writes:

For practical purposes, care would have to be exercised in the use of sliding tempo, in order to control the relation between tones in a sliding part with those in another part being played at the same time: a composer would have to know in other words, what tones in a part with rising tempo would be struck simultaneously with other tones in a part of, say, fixed tempo, and this from considerations of harmony. There would usually be no absolute coincidence, but the tones which would be struck at approximately the same time could be calculated.

### Phase Music

Can we calculate the times of coincidence like Cowell suggests? How? Can a musician even perform polytempic music with tempo accelerations or decelerations? Over the course of the 20th century, musicians explored this idea in what is now called phase music (wikipedia). One early example is Piano Phase (1967) by Steve Reich (wikipedia, vimeo).

Through experimentation, Steve Reich found that if the music is reasonably simple, two performers can make synchronized tempo adjustments relative to each other well enough to yield compelling results. In Piano Phase two pianos both play the same repeating 12 note musical pattern. One musician plays at a slightly faster rate than the other. It is possible for a musician to play this, but there is a limit to complexity of the patterns that a musician can play and a limit to the number of simultaneous changing tempi a musician can comprehend. Piano Phase approaches the limit of what a musician can perform. Can we not use electronic sequencers or digital audio tools to synthesize complex phase music with many simultaneous tempic transitions?

Most phase music (including Piano Phase) layers identical copies of musical content playing at slightly different speeds. The layers begin in-phase with with each other, and gradually drift out of phase. Reich's Piano Phase is no exception. As the layers drift out of phase, our listening attention is drawn to variations caused by shifting phase relationships between the layers. In the example below, the two layers start out playing together at 90 beats-per-minute (BPM), at $t=0$ the top layer abruptly starts playing at 100 BPM.

If we wait long enough, eventually the layers will re-align, and and the whole pattern will start from the beginning again. But what if we want to control exactly when the layers re-synchronize? Can we specify precisely both the time and the phase of sync points without abrupt changes in tempo? How can we sequence patterns like the one below exactly? This is the central question of Tempic Integrations.

### Tempic Integrations

If we want to specify the exact number of beats that elapse between the beginning of a tempo acceleration and the end point of the acceleration, and we want to hear this precisely we need a mathematical model for shaping our tempo acceleration curves. The goal is to compose and audition music where:

• Swarms of an arbitrary number of simultaneous tempi coexist.
• Each individual player within the swarm can continuously accelerate or decelerate individually, but also as a member of a cohesive whole.
• Each musical line can converge and diverge at explicit points. At each point of convergence the phase of the meter within the tempo can be set.

We start by defining a single tempo transition. Consider the following example from the image above:

• Assume we have 2 snare drum players. Both begin playing the same beat at 90 BPM in common time.
• One performer gradually accelerates relative to the other. We want to define a continuous tempo curve such that one drummer accelerates to 120 BPM.
• So far, we can easily accomplish this with a simple linear tempo acceleration. However, we want the tempo transition to complete exactly when both drummers are on a down-beat, so the combined effect is a 3 over 4 rhythmic pattern. Linear acceleration results in the transition completing at an arbitrary phase.
• We want the accelerating drummer to reach the new tempo after exactly 20 beats.
• We also want the acceleration to complete in exactly 16 beats of the original tempo, so that the drummer playing a constant tempo and the accelerating drummer are playing together.

We are interested in both the number of beats elapsed in the static tempo and in the changing tempo, as well as the absolute tempo. If we think of the number of beats elapsed as our position, and the tempo as our rate, we see how this resembles a physics problem. If we have a function that describes our tempo (or rate), we can integrate that function, and the result will tell us our number of beats elapsed (or position). Given the above considerations, our tempo curve is defined in terms of 5 constants:

• Time $t_0=0$, when the tempo transition begins
• A known time, $t_1$, when the tempo transition ends
• A known starting tempo: $\dot{x}_0$
• A known finishing tempo: $\dot{x}_1$
• The number of beats elapsed in the changing tempo between $t_0$ and $t_1$: $x_1$

The tension of the tempo curve determines how many beats elapse during the transition period. The curve is well-defined for some starting acceleration $a_0$ and finishing acceleration $a_1$, so we define the curve in terms of linear acceleration. Using Newtonian notation we can describe our tempo acceleration as:

$$$\ddot{x}_1 = a_0 + a_1t_1$$$

Integrating linear acceleration (above) yields a quadratic velocity curve (below). The velocity curve describes the tempo (in beats per minute) with respect to time.

$$$\dot{x}_1 = \dot{x}_0 + a_0t_1 + \frac{a_1t_1^2}{2}$$$

Integrating velocity (above) gives us a function describing position (the number of beats elapsed with respect to time, below).

$$$x_1 = x_0 + \dot{x}_0t_1 + \frac{a_0t_1^2}{2} + \frac{a_1t_1^3}{6}$$$

With equations for $\dot{x}_1$ and $x_1$, we can solve for our two unknowns, $a_0$ and $a_1$. First we solve both equations for $a_1$:

$$$a_1 = \frac{-2}{t_1^2}(\dot{x}_0-\dot{x}_1 + a_0t_1) = \frac{-6}{t_1^3}(\dot{x}_0t_1-x_1 + \frac{a_0t_1^2}{2})$$$

Assuming $t_1 \neq 0$, we solve this system of equations for $a_0$:

$$$a_0=\frac{6x_1-2t_1(\dot{x}_1+2\dot{x}_0)}{t_1^2}$$$

Evaluating for $a_0$ with our constants gives us our starting acceleration. Once we have $a_0$ we can solve our velocity equation to find the value of $a_1$. Plugging $a_1$ and $a_0$ into the velocity equation will give us velocity curve that produces a continuous BPM curve over the course of the tempo transition.

Note: We must specify the same time units for input variables like $t_1$ and $\dot{x}_1$. I prefer minutes for $t_1$ and beats per minute for $\dot{x}_1$ over seconds and beats per second.

In this post we solve the Fibonacci sequence using linear algebra. The Fibonacci equation is a second-order difference equation which is a particular type of sequence.

## Sequences

A sequence is a (possibly infinite) set of numbers with a defined order.

$a_n = \frac{1}{n}, \textit{ for } n = 0, 1, 2, ...,$

### Difference Equations

A difference equation is a type of sequence. A difference equation expresses the value of the sequence relative to the value of another point in the sequence.

$x_{n+1} = 1.5x_n, \textit{ for } n = 0, 1, 2, ...,$

To evaluate this equation at $n=3$ we need to know the initial value $x_0$, and evaluate $n=1, 2, 3$ in succession. For example, if $x_0=100$:

\begin{align} x_0 & && = 100 \\ x_1 & = 1.5 \times 100 && = 150\\ x_2 & = 1.5 \times 150 && = 225\\ x_3 & = 1.5 \times 225 && = 337.5 \end{align}

In the example above, we evaluate the equation recursively. It would be better to be able to evaluate the equation at $n = 100$ without performing $100$ calculations. Note that we can also write the equation above in this form:

$x_3 = 100 \times 1.5 \times 1.5 \times 1.5 = 337.5$

Solving a difference equation means writing the same equation such that it can be evaluated at a given $n$ without first evaluating $n=1,2,etc$. The solution to the equation above is:

$x_{n} = 1.5^{n} \times x_0, \textit{ for } n = 0, 1, 2, ...,$

### Second Order Difference Equations

The previous example is a first order difference equation because it expresses the value of $x_n$ as a function of one other sample, $x_{n-1}$. A second order difference equation is a function of two other samples, for example

$x_{n+2} = 1.5x_n + 1.4x_{n+1}, \textit{ for } n = 0, 1, 2, ...,$

## Fibonacci Numbers

The easiest way to describe the Fibonacci sequence is with a second order difference equation. Assume $F_0 = 0$ and $F_1 = 1$

$F_{k+2} = F_{k+1} + F_{k}$

The first eight Fibonacci numbers are:

$F_k=0,1,1,2,3,5,8,13,...$

We can solve the sequence with linear algebra. Let us start our sequence with a vector, $u_0=\begin{bmatrix}1\\0\end{bmatrix}$. Find the next vector in our sequence, $u_1$ like this:

$u_{1} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$

More generally:

$u_{k+1} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix} u_k$

Note that $u_0$ contains $\begin{bmatrix}F_1\\F_0\end{bmatrix}$.

The key to solving the sequence is in the eigenvalues and eigenvectors of $\begin{bmatrix}1&1\\1&0\end{bmatrix}$. We can write $u_3$ as

$u_3 = \begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix} \begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix} \begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix} \begin{bmatrix}1 \\ 0\end{bmatrix} = \begin{bmatrix}3 \\ 2\end{bmatrix} = \begin{bmatrix}F_4 \\ F_3\end{bmatrix}$

import numpy as np
fib = np.matrix([[1,1],[1,0]])
u_0 = np.matrix([[1],[0]])

fib * fib * fib * u_0 # The * operator is dot product for np.matrix (but not np.array)
matrix([[3],
[2]])

### Eigenvector Solution

The general case of the previous example is:

\begin{align} u_k &= \begin{bmatrix}1 & 1 \\ 1 & 0\end{bmatrix}^k\begin{bmatrix}1 \\ 0\end{bmatrix}\\ u_k &= A^ku_0 \end{align}

We can use eigenvectors and eigenvalues to solve the matrix power (see Eigenvalues and Eigenvectors). First, find the eigenvalues of $A$:

\begin{align} det\begin{vmatrix} 1 - \lambda & 1\\ 1 & - \lambda \end{vmatrix} & = 0\\ -\lambda(1-\lambda) - 1 & = 0\\ \lambda{}^2 - \lambda - 1 & = 0 \end{align}

Solve using the quadratic formula $\frac{-b \pm \sqrt{b^2-4ac}}{2a} \implies \frac{1 \pm \sqrt{5}}{2} \approx \frac{1 \pm 2.236}{2}\\$, which gives the eigenvalues for our matrix: \begin{align} \lambda_1 & \approx &1.618 \\ \lambda_2 & \approx &-0.618 \end{align}

Now we plug our constant eigenvalues ($\lambda_1$ and $\lambda_2$) into $(A-\lambda I)\vec{x}=0$ to find the eigen vectors.

$\begin{bmatrix}1-\lambda_1 & 1 \\ 1 & -\lambda_1 \end{bmatrix}\vec{x}_1 = 0 \implies \vec{x}_1 = \begin{bmatrix}\lambda_1 \\ 1\end{bmatrix}$

Note that this solution is not necessarily intuitive. This is not a valid solution for any arbitrary value of $\lambda$: our values, $\lambda_1$ and $\lambda_2$ are valid solutions. We can use the same process to solve for $\lambda_2$, revealing eigenvectors $\begin{bmatrix}\lambda_1 \\1\end{bmatrix}$ and $\begin{bmatrix}\lambda_2 \\1\end{bmatrix}$. Now create the eigenvector matrix, $S$ and eigenvalue matrix, $\Lambda$.

\begin{align} S &= \begin{bmatrix}\lambda_1 & \lambda_2 \\ 1 & 1\end{bmatrix}\\ \Lambda &= \begin{bmatrix}\lambda_1 & 0 \\ 0 & \lambda_2\end{bmatrix} \end{align}

Remeber that $A^k = S^{-1}\Lambda^kS$. To proceed, invert $S$. Remember this shortcut for inverting a $2 \times 2$ matrix.

\begin{align} \begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} =&& \frac{1}{ad-cb} &\begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \\ S^{-1} =&& &\begin{bmatrix}\lambda_1 & \lambda_2 \\ 1 & 1\end{bmatrix}^{-1} \\ S^{-1} =&& \frac{1}{\lambda_1-\lambda_2} &\begin{bmatrix}1 & -\lambda_2\\ -1 & \lambda_1 \end{bmatrix}\\ S^{-1} =&& \tfrac{1}{\sqrt{5}} &\begin{bmatrix}1 & -\lambda_2\\ -1 & \lambda_1 \end{bmatrix} \end{align}

Using the components $(A, \Lambda, S, S^{-1},u_0)$:

\begin{align} u_k &= A^ku_0 \\ u_k &= S\Lambda^kS^{-1}u_o \\ u_k &= S\Lambda^k\frac{1}{\sqrt{5}}\begin{bmatrix}1 & -\lambda_2\\ -1 & \lambda_1 \end{bmatrix}u_0 \\ u_k &= \frac{1}{\sqrt{5}}S\Lambda^k\begin{bmatrix}1 & -\lambda_2\\ -1 & \lambda_1 \end{bmatrix}u_0 \\ u_k &= \tfrac{1}{\sqrt{5}} \begin{bmatrix}\lambda_1 & \lambda_2 \\ 1 & 1\end{bmatrix} \begin{bmatrix}\lambda_1 & 0 \\ 0 & \lambda_2\end{bmatrix}^k \begin{bmatrix}1 & -\lambda_2\\ -1 & \lambda_1 \end{bmatrix} u_0 \\ u_k &= \tfrac{1}{\sqrt{5}} \begin{bmatrix}\lambda_1 & \lambda_2 \\ 1 & 1\end{bmatrix} \begin{bmatrix}\lambda_1^k & 0 \\ 0 & \lambda_2^k\end{bmatrix} \begin{bmatrix}1 & -\lambda_2\\ -1 & \lambda_1 \end{bmatrix} \begin{bmatrix}1\\0\end{bmatrix} \\ u_k &= \tfrac{1}{\sqrt{5}} \begin{bmatrix}\lambda_1 & \lambda_2 \\ 1 & 1\end{bmatrix} \begin{bmatrix}\lambda_1^k & 0 \\ 0 & \lambda_2^k\end{bmatrix} \begin{bmatrix}1 \\ -1 \end{bmatrix}\\ u_k &= \tfrac{1}{\sqrt{5}} \begin{bmatrix}\lambda_1 & \lambda_2 \\ 1 & 1\end{bmatrix} \begin{bmatrix}\lambda_1^k \\ -\lambda_2^k\end{bmatrix} \end{align}

## Verify Solution in Python

import numpy as np

lmbd_1 = (1+np.sqrt(5))/2
lmbd_2 = (1-np.sqrt(5))/2

a       = np.matrix([[1,      1],      [1,  0]])
s       = np.matrix([[lmbd_1, lmbd_2], [1,  1]])
s_inv   = np.matrix([[1,     -lmbd_2], [-1, lmbd_1]]) * (1/np.sqrt(5))
eig_val = np.matrix([[lmbd_1, 0],      [0,  lmbd_2]])
u_0     = np.matrix([[1],              [0]])

def eig_k(k):
return eig_val ** k

def u_k(k):
return s * eig_k(k) * s_inv * u_0

def fib(n):
f_n = u_k(n)[1,0]
return int(round(f_n))

[fib(n) for n in range(15)]
[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377]