[MUSIC] Welcome back. In today's lecture, I want to introduce an important mathematical construct called the Fibonacci Q-Matrix. And the Fibonacci Q-Matrix will be useful to us in deriving an identity called the Cassini's Identity. So how do we introduce the Q-Matrix? Let's go back to this table of Fibonacci rabbit pairs. So, remember in the first month we introduce one juvenile and then we have an adult in the second month. And then the adult gives birth to a new juvenile peer in the third month and so on. So we can look at the growth of the adults and the juveniles in this table. Of course if you look at the numbers you see 1, 1, 2, 3, 5, 8, 1, 1, 2, 3, 5, 8 in both rows. So we have the Fibonacci numbers there. But let's try to write down some equation for the coupling of the juvenile rabbits and the adult rabbits. So let's see how that works. So we need to define separately the number of juvenile rabbit pairs and the number of adult rabbit pairs. So let's start with the adults. So we have a sub n is going to be the number of adult rabbit pairs, right? And b sub n, then, we'll call the number of juvenile rabbit pairs. Okay, and we want to write then in a recursion relation for the number of adult rabbit pairs and number of juvenile rabbit pairs. So the number of adult rabbit pairs in the n plus one month, now where did they come from? From the nth month. Well, an adult rabbit so let's say look at this the number here 3 where did this 3 come from? It came from the two adults in the previous month, because no rabbits die in this Fibonacci population. And it also came from this juvenile up here, because the juveniles after one month mature into adults. So the adults in the n + 1 month comes from the adults in the nth month plus the juveniles in the nth month. Right? Okay. What about the juveniles in the n + 1 month? Where do they come from? So let's say these juveniles here, where did they come from? Well, they came from these adults giving birth, right? The females in this adult rabbit pair giving birth. So it came from the number of adults in the previous month. So the number of juveniles in the n + 1 month is equal to the number of adults in the nth month, okay? This is a matrix equation, right? We can write this, so an plus 1, bn plus 1. So we put that as a column vector and that's equal to a matrix times a sub n the number of adults in the end month. And b sub n, the number of juveniles in the nth month. So what is the relationship? A sub n plus one equals a sub n plus b sub n. So we have a 1 1 here. And b sub n plus 1 equals a sub n. So we have a 1, 0. Okay? This matrix here is what's called the Fibonacci Q matrix. Okay? So how does the Fibonacci Q matrix work? The Q matrix brings the The Q matrix brings the population from the nth month to the n plus 1 month, okay? That's what the Q matrix does. So let's look at that, so then if we want the population say in the n plus k month, all right. So we want to move the population k month forward. We need to multiply k times. So we need to multiply Q to the K times the population in the nth month. Okay? So, this Q matrix is the matrix that moves the population from one generation to the next generation. And if we want to move the population K generations, we need to multiply the population in the nth month by Q, k times, okay? So here I want to see what is this thing, right? What is Q to the K power? So lets have a look at this. We have Q, right, Q is our 1, 1, 1, 0 matrix. So, we want to find what Q to the K is. To figure out what Q to the K is, we can see what this Q do when we multiply a matrix by Q. So let's take Q and multiply it against some 2 by 2 matrix. So A, B, C, D. So what does Q do to this matrix? Well, you multiply across the row and down the column. So 1 1 against ac. So the first row will become a plus c. And then 1 1 against the b d and the second element here will be b plus d. And then you do the second row, so multiply 1, 0 against a, c will just be a. And 1, 0 against b, d will just be b, okay? So multiplication of a 2 by 2 matrix by Q replaces the first row by the sum of the first row and the second row, and replaces the second row by the first row. Okay? Replaces the first row by the sum of the two rows, and the second row by the first row. So let's see how we can use that. Right? So Q is now a 1, 1, 1, 0. We want to figure out what Q squared is, Q cubed is, etc. So we're going to be replacing, as we multiply Q by Q, we're going to be adding the first and the second row. And replacing the first row by that. But we know that Fibonacci numbers have this nice recursion relation. When you add two Fibonacci numbers in a row, you get the next Fibonacci number. So these 1's are Fibonacci numbers. F1 Is a Fibonacci number, right, which is equal to 1. And F2 is a Fibonacci number, which is equal to 1 and F0 is a Fibonacci number, which is equal to 0. So, actually this 0 is a unique Fibonacci number, which is F0 and then we're going to be adding, when we multiply Q by itself, we're going to be adding the first row to the second row, right? To replace the first row. So we can add two Fibonacci numbers here to get another Fibonacci number. So F1 is 1 so we can replace 1 by F1 and then the same thing here, we can make this an F1. And F2 is also 1. So we can replace this top one by F2, okay. So we can write the Q matrix as a matrix of Fibonacci numbers. Then when we determine Q squared, remember we replace the first row by the sum of the first row in the second row. So F1 plus F2, we can use the recursion relation and we get F3. And F0 plus F1, we can use the recursion relation and we get F2. And the second row gets replaced by the first row. So then we have an F2 and then F1, okay. And it continues like this. So Q cubed will replace the first row by the sum of the first row and the second row. So this will become an F4. And we'll replace this element here by the sum,, so this will become an F3, and so on and it will build up. So I think at this point we can write down the general result. So Q to the nth power. This lower element here matches the power. So this is F sub n. This corner element up here matches the power. So this is F sub n. And then F1 is one lower. So this is Fn minus 1. And F3 is one higher. So this is Fn plus 1, okay? And that's the general result for the Fibonacci Q matrix raised to the nth power, okay? This will be an important result for us because it will make the proof of a theorem, which is called Cassini's Identity, rather elementary, okay? So I'll see you next time.
