Matrix Exponentials, Jordan Form & Differential Equations

Based on the supplied transcript. Punctuation, paragraph breaks, spelling, and obvious name transcriptions have been cleaned up; repeated fillers and false starts have been reduced for readability. The speaker’s commentary and opinions are retained.

All right. This is the lesson recap on Volume Two, Chapters 8 to 12, Differential Equations and Linear Algebra, by Grant E. Gustafson. Gustafson? Gustavo Fring? Gus—Gustafson? Gustafson. Sorry if I mispronounce your name. I am a master linguist, but not a—what’s a pronouncer? Speaker? Dialect? Gustafson. Grant Gustafson. An ancient book from the archaic archive of intellect.

Yesterday, we spent two hours doing a problem from systems of differential equations. We chose to do what you guys might think is a proof. It’s not a proof. To verify or show means this side of the equation equals this side of the equation. Verify that it is true.

This is a book using software called Maple. Maybe that gave birth to MATLAB or something. I don’t know. It’s an old book from the University of Utah, I think, written in black and white and written very advanced. The person who wrote that book knew how to use TeXShop and LaTeX code very well. For those of you interested in why you might use TeXShop as opposed to Microsoft Word, it is because of all the citations, references, equations, and theorems.

It is a freezing one. Winter is upon us. It’s time to watch Game of Thrones. You should look at that book. You kids out there still don’t know what software is. This guy must have spent at least 30 years writing this book. It would have taken a long time, especially in the ’70s, ’80s, or ’90s—whenever he wrote it. It had to have been in the ’90s because TeXShop and LaTeX code didn’t come out until the ’90s, I think. Then Microsoft Word introduced it into its software in the early 2000s. I have to double-check that.

To type a book that large in that kind of code is like talking to Beethoven. There are great composers. Not everybody likes everybody’s music. Sometimes you learn to love everybody’s music. That’s something I’ve learned to appreciate about mathematical literature, particularly if you’re a student complaining about the textbook. It’s not the book. It’s never the book. It’s the reader, always, every time. You’re unpracticed in reading these books, which is the objective of my research program.

This is a two-part book, Volume One and Volume Two, and we went to the very end. If you’re new to Math Monthly, what we do in the research position is take random books we find on the internet and random problems from random chapters, and solve them using the book, the book, and only the book. We use technology for arithmetic at a very basic level and to try to help find a typo without looking at solutions.

In a recap of what we used AI for and what we avoided AI for, there was a time I didn’t know what Maple was. This is a book for engineers. Maple might have been software you needed to have on your résumé. It might have been required to live in Canada.

Here’s the thing about the research program. If you’re a math, physics, or engineering student and had gone through that with me, now you can put familiarity with Maple on your résumé. Those at yesterday’s lesson may not know how powerful what you experienced was. You can say, “I was in the research program with Math Monthly during this period, operating under Professor Jonathan D. Tullis, and I learned how to use Maple.” We don’t have Maple. It was a fun trip. We time traveled. It was the most exciting thing I’ve experienced in a long time, and we used AI to help us time travel.

We took this question. You all know e to the at, right? How about e to the capital At? Euler’s number, e—it’s not necessarily a number; it’s like pi, but it’s a summation of an infinite amount of numbers.

I found something in that book. This is the thing about research that we’re doing. For those who attended—it wasn’t a lesson; we’re doing research together—I found something I’ve been searching for for 15 years. Not actively looking for it, but something in the back of my mind. I can’t remember whether I proved it a different way. I proved it in one way; I can’t remember.

You have e, which is the summation of one over n factorial from zero to infinity. If you add an exponent, you’re going into power series. You squeeze in the power. If it’s e to the at, then it’s the summation from zero to infinity of a to the n, t to the n, over n factorial—or k, whatever letter you’re using.

If you turn that into a capital A, meaning a matrix, you have e, which is a constant, raised to a matrix. Imagine two raised to a square matrix. It may not make sense to you, but that’s because you don’t understand math yet. You’re still thinking math is arithmetic and physics. When somebody asks what one times zero equals, you still talk about apples. We’re not in physics. We’re in theory.

Theoretically speaking, why does one times zero equal zero? I know you want to answer that, but there’s no answer to that. The answer would be: according to what source and what rules are you talking about? There’s no magic to it. To go into detail about why one times zero equals zero would mean you’re an idiot. I’ve heard people tell me about famous people talking about that. I’m like, “Yeah, they’re idiots. You’re listening to idiots on the internet. Have you not figured that out yet?” Why do you think I deleted everything? I deleted over 10,000 videos because I was an idiot. I woke up. I’m trying to wake you up. I am Morpheus. Where is Neo? Better yet, where’s the lady in the red dress? I need someone to match my outfit.

One special thing we discovered in this research process was in this ancient, archaic book that looks like it was written on a typewriter, probably with big, fat keys and a black-and-green screen. You have no idea how much code. When you look at a mathematically based textbook, the code condenses when you compile it, output it, parse it, or whatever.

What I found was the proof that e to a matrix multiplied by t equals the summation of A to the n, t to the n, over n factorial. That’s an identity we all know and use. It’s fascinating because, although it’s a matrix, it operates almost identically to a constant. You take a derivative, and it does the same thing in the matrix world for solving systems of differential equations. That proof is amazing. It was God speaking to me.

This is what research is, kids. We’re doing research because we need to expand our minds, our database. What you learn in college is a precursor. What you do outside college holds true value.

Video games, OJ, and Sunny D or whatever are not going to—you’ve got to change your diet, too. There’s a reason why I look like this at 42. I’ve been cooking for myself from scratch since 12. I’ve been eating things like kale since before people knew what kale was. I went to a consignment—concessions, consignment, whatever—a place selling used goods and bought one of those ancient juicers from the ’70s. I’ve been juicing since I was like 10. I did this by my own choice. My family was eating fast food. I was dying. My body requires a superior fuel source.

If you’re in this field, recognize that your brain is like a very advanced machine and requires very advanced fuel regularly. I probably eat thousands of different ingredients every week: spices, vinegars. I ferment foods. That’s why I do the cooking show. It shows you how to maintain the brain. My cooking show is for scientists.

You don’t understand the touching of food, smelling food, the artistic process. That’s very important. Your body absorbs enzymes and things from touching the vegetables you’re chopping. The fumes you’re smelling, just like smoking something, are intoxicating. They have an effect. If you can take a hit of tobacco or marijuana and have that effect, don’t you think food fumes would have an effect as well? You should cook for yourself. It’s natural for people like us to want to write literature, cook, and listen to music. There are no ads in it.

For those who were there, what I took away was that proof. They used the Picard iterate to prove it. I remember first seeing the Picard iterate and thinking, “What is this gibberish?” Then I saw this proof and thought, “Oh, I’ve been looking for this proof for like 15 years.” That’s research. Found it.

Looking at that proof adds to my library of opportunities and choices. When I look at a proof that might stump me, it gives me a wider variety of tools and a broader spectrum of angles, approaches, and ways of looking at things.

For the LaTeX code, I type everything in Microsoft Word. But if you were typing a scholarly paper with 20,000 pages of code and needed hyperlinks—if you were in the lesson, you watched me click hyperlinks to go back to the theorems. Boom, boom, boom. That’s why you would use TeXShop: for a super-long scholarly paper. Those going into industry would never use that, for the most part. You would use something like Microsoft Word, far more efficient. It uses the exact same code, LaTeX, in the exact same way. It just has a different casing.

I’ll show that and do a review. This is the morning review of yesterday’s research position for those taking part. I want to tell you the highlights I took from it. The proof for the exponential with the matrix was a very valuable thing for me to see. It made me happy.

We went through this and learned about some old software, Maple. There was another one—Mathematica, I think. Is that still used? I don’t know. MATLAB is. There are so many, and they’re always changing.

With Maple, we took a drive and time traveled into the past. We were doing this question. Most of you would have been given this equation and tried to solve it like a [inaudible]. We did it the way it’s supposed to be done, following the instructions. We backtracked through everything and found the theorems and everything we needed: Jordan form. There were all these Jordan forms. I’ll have to refresh. Any of you who have taken linear algebra have heard “diagonalization.” The Jordan form of a matrix. It was diagonalization, although it was talking specifically about a diagonal matrix, which separates the titles.

We backtracked to find this piece, then this piece, then this piece. Halfway through the show, it wanted us to use this old-ass software and gave us the code. We had to retype the code with our information. Easy to do. Plug a couple numbers in and type the code into Maple.

That, my dear boy, girl, t-boy, t-girl—that, my little freak—is when you can use ChatGPT. There’s no difference between dropping that code into Maple, software I don’t have, and ChatGPT and having it pretend to be Maple. I prompted it to be Maple and give us that experience. I took the screenshot and put it in.

We learned how to use Maple. We have experience with Maple now. You can add familiarity with Maple to your résumé. If there’s a 60-year-old interviewing you and they see that, they’re going to say, “You’re hired. I don’t even know people my age who know what that is.” This is the value I’m offering.

Familiarity, not expertise. You know how to drop code into a thing. That’s the point of doing research with people like me: adding that stuff to your résumé. It’s not about being an expert. It’s about understanding how to learn quickly. When you get to the workforce, they’ll want to know you can pick up new things quickly and aren’t scared when you see new things, which means you need exposure to a lot of things.

We time traveled. It was awesome. We did some mathematical time traveling. Then we got our answer and continued through the process: a long, elaborate process with lots of broken pieces of arithmetic to find the matrix.

Most of you probably would have taken that equation, if you knew the exponential relation, and diagonalized the matrix. That would have been wrong. You would have gotten zero credit and complained, “I got the answer right.” The professor would say, “Where in the book does it say you can use diagonalization?” In your head: “But I’m an egotistical, arrogant little freak. I’m right. My mom still wipes my ass. Of course I’m right.” You kids need to leave.

We don’t leave the textbook. We’re not out of college; we’re in college. You don’t leave the textbook. It defeats the whole purpose of college. That’s what I’m showing you. Most listening will never succeed in this because you can’t even listen. You already stopped listening like 20 minutes ago.

We spent two hours doing this problem and still haven’t finished because we made a typo. We did it correctly. We got to the end and got the right answer, but there was a typo. Such a silly mistake, because I’m an egotistical, arrogant, pompous little prick, too. I did a two-by-two determinant adjugate on an upper triangular, non-fractional matrix. All I had to do was add a negative sign. Instead, I flipped the diagonal. I flipped both diagonals.

All it was was one, one, zero, one. All I had to do was add a negative sign, and I knew that. I can make excuses: it was late at night; I’m tired. There are no excuses. Get out of here. You’re fired. You just blew up a rocket because you thought you could do arithmetic in your head. That’s what I did.

These are things you’re supposed to experience in college so you don’t do them in the workforce. Also, not getting angry, staying calm, and not getting frustrated when you can’t solve the problem are very important. Most of you might have a mental breakdown. If you can’t control that, you’re in the wrong place.

We looked at it and went back. I didn’t write the formula out, which is a huge mistake. You’ve got to write the formula out. I didn’t do what I tell you to do or what you’re supposed to do because it was so minute that I assumed. That’s what you do on your exams. In those minute moments, you assume because you think you’re smart.

Let me explain something, kid. Anyone studying math, physics, or engineering is a janitor. You’re not smart. You’re a janitor. You’re going to work 12 hours a day, every day, doing math and physics for the rest of your life. You’ll be paid handsomely, but you’ll never spend it. All your work is for the benefit of others. If you can’t understand that, you’re in the wrong place.

We found the typo. The process was correct. It was so unique. The solution you do compared with what you think you would have done is so unique. This is a book for engineers, which is why coding was incorporated. We went through the engineering process of solving this with the assistance of coding software. A mathematician might use a different approach or software. A physicist might do it differently or use different software. It depends. There are always many ways to solve these problems.

At the end, I told you to complete it. You have the book. As part of the research program, you now know where it is. We find books freely available on the internet. We do not pirate digital content. A person who pirates digital content is a felon. You’re committing white-collar crime. Most people don’t think that’s an actual crime. It’s the worst kind, a coward’s crime. I have respect for people in prison who rob people. I mean, they suck, but at least they’ve got balls to rob somebody. You have to hide in a closet and click a button. It’s pathetic.

We find freely available books and take whatever pops up, however old, weird, or crazy. We go through it, learn how to use it, and learn new things. I learned a lot yesterday. I could have solved that problem in two seconds using diagonalization and dropping the matrix into WolframAlpha to find the eigenvalues, eigenvectors, and eigenspace. I could do it myself or drop it into Wolfram: two seconds, solve it, and walk away.

Then I’d be completely useless. I would have learned nothing and exercised nothing. That mentality you have with AI—we used AI correctly to travel back in time to ancient software. It was fun. In my mind, we were on an old computer typing on a black-and-yellow screen with old tube monitors, the stuff they incubated you in.

It was a good lesson, a great lesson. It was poetic. I don’t know whether I’ll complete it because it’s completed. The journal is supposed to be interactive. We did the solution correctly. I just made a typo.

Do you know the Hubble telescope? This is a story you might not know. When NASA, the people who built that billion-dollar telescope or something like that, put it in space, they put the wrong lens in it. They took a picture, and it was blurry. It’s in space now, and they put the wrong lens in it.

The head of NASA gets a call: “You put the wrong lens in that telescope.” This would be like Oppenheimer getting a call: “You put the wrong uranium in the bomb.” The head of NASA is instantaneously like a teenage boy: “I have four PhDs, with five PhDs working under me, and 50,000—we—there’s absolutely no way.” Then he looked: “They put the wrong lens in there.”

I made that typo. We went back and solved it. Very minuscule, not even a speck of dust compared to that. But that’s the point: stepping stones of experiencing something like it, because in the workforce you’ll make mistakes. If you get stressed and frustrated—“I did all this work. Now I have to redo it”—you’re in the wrong place if you can’t change your mentality.

Keep yourself calm. It doesn’t bother me anymore. I like it. It’s like, “Now I know where my mistake was, and I won’t do that again.” Although I’ve said that before at the basic arithmetic level. Now I’m like, “I’ll not do that again.” Don’t tell me to make a mistake. You’re going to make mistakes. We find the mistake.

I don’t know if I’m going back through it because the solution’s done. We did it correctly. It’s clearly correct. There’s just a typo, so it’s completely wrong. The solution’s wrong. The answer’s wrong. There’s a typo.

The journal comes to you, and I put this on my website. You can go and type in traditional LaTeX code. All packages are on my website—or should be, at least. You can interact with each other and solve the problem. I’ll show you how. You can work together to complete and verify it. Upload your own typed image or document, or type directly into the comment form and respond interactively.

That’s part of the research program: interacting after the session. There’s no reason for me to go back through it. We know the solution. In the real world, you’d use technology to help verify and avoid typos, but that’s not the point. The practice is to find things we’ve never seen and become familiar with things people 30 years before us knew. Now, in conversation with another professor, when somebody says, “Remember Maple?” I can say, “I do remember Maple.” It’ll increase your conversability.

This is Math Monthly. I do a monthly journal. I don’t have a copy right now. I’ve got to print it, cut it up, and put it together. I make and manufacture them here. They’re awesome. You’ve seen them. They look like these, just thinner. These are my manufactured books. This is my crash course, The Ultimate Crash Course for STEM Majors. The journal looks like this, about half the size, with a different cover every month.

Those interested in expanding your mind can join Math Monthly and become superior. If you join, I’m not here to answer your questions. If you have questions, go to your professor. Read your code of conduct and syllabus. Answer your own damn questions. We’re doing research. We’re here to engage in professionalism and enhance our knowledge bank and skill set. I’m not here to answer the questions of a whiny little baby.

That’s my morning coffee recap. Am I allowed to coffee? One last sip. That’s my morning recap of yesterday’s session. If you’d like to join the party, join the party. The aftermath—the afterparty? I put this on the website, and you can go in and learn how to type. More things for your résumé: HTML, web development, interaction, and communication skills. Written and oral communication are number one on the résumé. Nobody wants to hire an illiterate person who cannot speak to somebody.

If you’re a shy young man watching this, it’s time to buck up, boy. You’ll have to talk with people like me. We have no sympathy for your family values. You’ve got to grow up. Communication is number one. Some of the most brilliant people on the planet still live at their parents’ house doing nothing because they cannot talk to somebody.

If you want to overcome that fear, go to a gym, a male locker room. Take off your clothes and stand naked with the rest of the men. Take a sneak peek at everything. Watch the old dudes bend over. Walk around. Nobody cares. Learn that you’re not the center of the universe. That’s your problem. You need to overcome it.

If you’re a non-native English speaker, I highly recommend learning the dialect of English if you’re coming to America. It’s very difficult for Americans to listen to foreign dialect. We’re trained in the proper way to speak. That’s good old-fashioned advice for anybody wanting to be in America. America is the land of a racist. It’s more like discrimination, actually. If you don’t want somebody to be discriminatory toward you, you must speak with the proper dialect. It’s okay to have an accent, but speak clearly so people understand what you’re saying, with an accent. Understand?

All right. Today, we’ll see what’s on the menu. Yesterday was linear algebra with differential equations for engineers. Have a nice day.

This recap reflects on a roughly two-hour Math Monthly session using a problem from Grant E. Gustafson’s Differential Equations and Linear Algebra. The emphasis was on reconstructing a textbook’s method: locating definitions, tracing referenced theorems, interpreting code, and checking the resulting calculations.

Matrix Exponentials & Picard Iteration

Professor Tullis identifies a proof involving Picard iteration as the session’s most valuable discovery. It connected a familiar matrix-exponential series with a method of successive approximation, expanding the range of proof techniques he could draw on in future work.

Textbook Methods & Jordan Form

The discussion emphasizes following the problem’s requested method and tracing its supporting material rather than immediately choosing a familiar shortcut. Jordan form and diagonalization appear in that discussion. For mathematical clarity, diagonalization is the special case of Jordan form in which every Jordan block has size one; a general Jordan form can include larger blocks.

Maple Code & Simulated Output

The textbook incorporated Maple code. Because Maple was unavailable during the session, ChatGPT was asked to imitate its output. This provided exposure to the code and its intended workflow. Simulated output is not a verified Maple computation, and reviewing it is different from operating Maple directly.

Finding & Correcting a Matrix Error

An error in a two-by-two matrix inverse affected the final calculation. The recap uses that mistake to emphasize writing out formulas, checking arithmetic, and staying calm while tracing an error. A sound overall approach still requires correct calculations and a final verification.

Independent Review & Collaboration

Participants are encouraged to revisit the journal, correct the calculation, verify the result, and discuss their work through the website. The broader goal is to develop mathematical reading, technical communication, familiarity with unfamiliar tools, and the ability to learn new methods.

This summary focuses on the mathematical and research-related content. Personal anecdotes and broader commentary remain in the transcript above.