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PROFESSOR: The previous video was about positive definite matrices. This video is also linear algebra, a very interesting way to break up a matrix called the singular value decomposition. And everybody says SVD for singular value decomposition. And what is that factoring? What are the three pieces of the SVD? So this is the fact is every matrix, rectangular, every matrix factors into-- these are the three pieces. U sigma V transpose. People use those letters for the three factors. The factor U is an orthogonal matrix, an orthogonal matrix. The factor sigma in the middle is a diagonal matrix. The factor V transpose on the right is also an orthogonal matrix. So I have orthogonal, diagonal, orthogonal, or physically, rotation, stretching, rotation. Now we have seen three factors for a matrix, V, lambda, V inverse. What's the difference? What's the difference between this SVD, this, and the V, lambda, V transpose, V inverse, V lambda, V inverse for diagonalizing other...

EMMANUEL KASIGAZI: What was that moment when you knew that OCW are the people who, you know? MARIA BARBOSA: You watch five minutes of Gilbert Strang, it is impossible not to be impressed! [EMMANUEL laughing. MUSIC PLAYING] EMMANUEL KASIGAZI: Welcome to Open Learners. MICHAEL PILGREEN: A podcast that tells the stories of learners all over the world who use MIT's OpenCourseWare. EMMANUEL KASIGAZI: I'm Emmanuel Olimi Kasigazi, an Open learner myself, from Kampala, Uganda, in east Africa. MICHAEL PILGREEN: And I'm Michael Jordan Pilgreen, an Open learner from Memphis, Tennessee. [MUSIC PLAYING] EMMANUEL KASIGAZI: Hi, Michael. MICHAEL PILGREEN: Hey, Emmanuel. EMMANUEL KASIGAZI: How you doing? MICHAEL PILGREEN: I'm doing great, man. I can't believe we're here. EMMANUEL KASIGAZI: Yeah can't believe we're actually recording this. This is something we've been talking about for a while now, I think over 13, 14 months. Do you remember the first co...

PROFESSOR: Well, yesterday was easy. You learned all of the rules of programming and lived. Almost all of them. And so at this point, you're now certified programmers-- it says. However, I suppose what we did is we, aah, sort of got you a little bit of into an easy state. Here, you still believe it's possible that this might be programming in BASIC or Pascal with just a funny syntax. Today, that illusion-- or you can no longer support that belief. What we're going to do today is going to completely smash that. So let's start out by writing a few programs on the blackboard that have a lot in common with each other. What we're going to do is try to make them abstractions that are not ones that are easy to make in most languages. Let's start with some very simple ones that you can make in most languages. Supposing I want to write the mathematical expression which adds up a bunch of integers. So if I wanted to write down and say the sum from i equal a ...

PROFESSOR: Well, yesterday was easy. You learned all of the rules of programming and lived. Almost all of them. And so at this point, you're now certified programmers-- it says. However, I suppose what we did is we, aah, sort of got you a little bit of into an easy state. Here, you still believe it's possible that this might be programming in BASIC or Pascal with just a funny syntax. Today, that illusion-- or you can no longer support that belief. What we're going to do today is going to completely smash that. So let's start out by writing a few programs on the blackboard that have a lot in common with each other. What we're going to do is try to make them abstractions that are not ones that are easy to make in most languages. Let's start with some very simple ones that you can make in most languages. Supposing I want to write the mathematical expression which adds up a bunch of integers. So if I wanted to write down and say the sum from i equal a ...