In this video, we will learn some of the important applications of diagonalizability, like power of a matrix, exponential of a matrix, etc. How can we compute power of a matrix? Now, the question arises, why diagonalizability? What's the need of studying diagonalizability? The first important applications of diagonalizable matrices is that we can compute power of a matrix by simply multiplication of three matrices. How it is possible? Let us try to understand in this video. Suppose A is a diagonal matrix of order n, then there exists an invertible matrix P such that A = PDP inverse where D is a diagonal matrix. Now, how we can compute the kth power of A. Suppose you want to find A squared. Very simple, A squared. What is A squared? A squared will be A into A. What is A? If it is diagonalizable, it will be PDP inverse. And it is also PDP inverse We can write it like this, PD P inverse P into D into P inverse. So, this is what P inverse P is identity and D into D is D squared. It is PD squared P inverse. Similarly, if I want to compute A cube, what will be A cube? A square into A, which is A squared is what? PD squared P inverse. Into A is what? PDP inverse. It is equal to PD squared into P inverse P. We can write in one bracket, D into P inverse, which is PD cube P inverse, since P inverse P. Similarly, if suppose you want to find out A raised to power 100, that will be what? That will simply equal to PD raised t power 100 into P inverse. Initiative multiplying A 100 times, you are simply multiplying these three matrices to compute A raised to power 100. This is happening only when A is diagonalizable. Now, how to compute the D raised to power 100 or D raised to power k, so D is what? D is a diagonal matrix. It is Lambda_1, 0, 0 0, Lambda_2, 0, and 0, 0, Lambda_m. If we want to compute D raised to power k. This is simply equal to Lambda_1 raised to power k, 0, and so on, 0 Lambda_2 raised to power k and so on, and 0, Lambda_n raised to power k. Let us try to find A raised to power 6 for this 2X2 matrix. Now, one can easily verify that this matrix is diagonalizable, because there exists an invertible matrix P, which is this matrix, such that A can be expressed as PDP inverse. Now, to compute A raised to power 6, as we have already seen that A raised to power 6 is simply PD raised to power 6 into P inverse. P is in. D raised to power 6 will be simply 1 raised to power 6 is 1 and 4 raised to power 6 will come here. This matrix will remain same. Now, the multiplication of these three matrices, which is this matrix, will simply give A raised to power 6. Now, another important application of diagonalizability is exponential of a matrix. If we want to find out e raised to power A where A is a matrix, how can you do that? This is doable if matrix A is diagonalizable. How? Let us try to understand it. Since e raised to power A can be written in the series form as identity matrix of order n cross n plus e + A squared upon factorial 2, A cubed factorial 3, and so on. This is by the series expansion of e raised to power A. Which is further written as I remain I, A is can be written as PDP inverse because A is a diagonalizable matrix. A squared can be written as PD squared P inverse, and similarly A cube and so on. Now, I can be written as P into P inverse. This entire expression can be expressed in this way. You can take P here and P inverse to the right of this. The inside expression is I +D + D square upon factorial 2 D cubed factorial 3, and so on, which is nothing but e raised to power D. So e raised to power A, which is this series expansion. This can be written as a matrix multiplication of only these three matrices. Now, how to compute e raised to power D here, e raised to power D is what? E raised to power D is I + D upon factorial 1, plus D squared upon factorial 2, +D cube upon factorial 3, and so on. I is identity matrix order n cross n. It is 1, 0, 0, and so on,0 0, 1, 0, 0 and 0, 0, and 1. +1 by factorial 1 times, this is diagonal matrix, which consisting of eigenvalues of A. It is Lambda_1, 0, 0, 0 0, Lambda_2, 0, and it is 0 up to Lambda _n. The next is 1 by factorial 2, it is D squared, which is Lambda_1 squared, 0, 0, 0 0, Lambda_2, squared and 0 0 up to Lambda_n squared and so on. Now, when you take the addition of these all matrices, you will get the first element as 1 by 1 +Lambda_1 upon factorial 1 plus Lambda_1 is square upon factorial 2 and so on. 0, 0 and so on 0. 0, 1 +Lambda_2 upon factorial 1, Lambda_2 squared upon factorial 2 and so on, and 0, 0, 0. Similarly, the last term will be, 1 +Lambda_n upon factorial 1 +Lambda_n squared upon factorial 2, and so on. This is nothing but e raised to power Lambda_1, 0, 0, 0. 0, e raised to power Lambda_2, 0, 0, 0, and 0, e raised to power Lambda_n. This is basically your e raised to power D. This is how you can compute e raised to power D if you want to find out e raised to power A over a given matrix. Now, let us take this example of order 2 cross 2 and you want to compute e raised to power A, in this case. This matrix is diagonalizable again, and we can find invertible matrix P such that the A can be expressed as PDP inverse. What is P? P is this matrix. To compute e raised to power A as we have seen, that we can return e raised to power A as P, e raised to power D into P inverse. What is P? P will remain same. P is this matrix, will remain same. E raised to power D will be e squared, e of Lambda_1 and e of e raised to power Lambda_2. P inverse will remain same. When you multiply these three matrices, you will get e raised to power A. Similarly, if you want to compute sin of A, sin of a matrix, you can simply expand by sin series and you can find sin of a matrix 2. In this video, we have seen some of the important applications of diagonalizability. We have seen how we can find out powerful matrix or exponential of a matrix.
