In this video, we will learn what diagonalizable matrixes are. We will also discuss two examples based on this. What are diagonalizable matrixes? Let A be a matrix of order n cross n. Then A is said to be diagonalizable matrix. If it is similar to a diagonal matrix D, that is, there exist a non-singular matrix P such that a can be expressed as PDP right here. When we can see the dramatic diagonalizable, if it is similar to a diagonal matrix D. Since eigenvalues, of those two similar matrixes are same, and the eigenvalues on the diagonal matrix are nothing but a diagonal elements. We can easily see that the diagonal matrix is nothing but having the diagonal entries as the eigenvalues of the matrix A. I want to say this. Suppose the matrix A has eigenvalues lambda one, lambda two, and so on up to lambda n. Then this diagonal matrix will be lambda 10000, lambda 200 upto lambda. This is become diagonal matrix D. Now, take this example. For this particular problem, we have to show that the matrix A is diagonalizable matrix. How can we show? For showing that it's a diagonalizable matrix, first of all, we have to find these eigenvalues. Eigenvalues will give the diagonal matrix D. Then we have to show the existence of a matrix P such that, AP equal to PD and P must be an invertible matrix. First of all, find its eigenvalues. How can you find the eigenvalues? We will write this characteristic polynomial, a minus lambda, I, put it equal to 0 . This implies determinant of six minus lambda minus one, two, three minus lambda equal to 0 . This implies six minus lambda into three minus lambda plus two equal to 0 , or this is lambda squared. This is minus nine lambda plus 80 plus two equal to 0 . That means lambda squared minus nine lambda plus 20 equal to 0 , which is lambda minus five into lambda minus four equal to 0 , or lambda equal to five and four. This means the eigenvalues of this matrix A are five and four. What is the diagonal matrix then? Diagonal matrix will be five 0 , 0 , four. Now, in order to show it's a diagonal matrix, we have to find a matrix P invertible matrix P, such that AP is equal to PD. Suppose this matrix P is a, b, c, d. Let us take AP equal to PD. This implies what; a is, what? Six minus one, two, three. P is what? a, b, c, d, which is equal to P, is a, b, c, d. D is the diagonal matrix which is five, 0 , 0 . This implies 6a-c, This should be equals to 5a and implies a equal to C. The first element of the left-hand side should be equal to first element of the right-hand side and similarly others. Now next is 6b - d should be equal to 4d, and this implies d equal to 2b. The third element is 2a plus 3c. That should be equals to 5c. This implies a is equal to C. So last entry is 2b plus 3d, is equal to 4d. That means, 2b equal to d, which is same as this. What will be p? It is a, b is b, c is a, because a is equal to C and d is 2b. Any such type of matrix will satisfy this equation, AP equal to PD. Now you can arbitrary put any value of a and b. Suppose you take a as one and b as one, one, one, one, two. This matrix is non-singular because determinant is non-zero . AP equal to PD which is also satisfying. We can say that, the matrix a is diagonalizable matrix. Now the question arises, is every square matrix diagonalizable? The answer is no. We have this example. You take this matrix a, then you can easily verify that both the eigenvalues of this matrix are one. There does not exist any similar matrix. Non-singular matrix P such that AP equal to PD. Why it is true? Because if you take P as a, b, c, d for example. What is a here is one, one, 0 , one. B is a, b, c, d. This is a, b, c, d. What is d? D is one, 0 , 0 , one. Diagonal matrix. This implies a plus c, this is b plus d, this is c, this is d. This must be equal to a, b, c, d, which further implies a plus c equal to a, or c equal to 0 . b plus d should be equal to b. This implies d equal to 0 , c is equal to C and D is equal to d is always hold. What would be P then? P will be a, b 0 , 0 . Any matrix of this type will satisfy this equation. But this P is a similar matrix not a non-singular matrix because determinant of this P is 0 . We can say that if a square matrix is given to you, then that matrix may or may not be diagonalizable. In this video, we have seen that what diagonalizable matrix are. We have also seen that if a square matrix is given to you, then the square matrix may or may not be diagonalizable.
