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Guys weve made it were were on the topic of diagonalization. So so you you got to be able to pronounce it i mean that thats one so diagonalization diagonalizability all these things work on it and then so yeah go pause the video and come back when youre confident in your pronunciation. So now we can like talk about what it actually is so first things first.
What is a diagonal matrix heres an example one zero zero. Two this is a diagonal matrix why because it has nonzero entries along the main diagonal and then zeros everywhere else and an interesting property of diagonal matrices is if you raise the diagonal matrix to some arbitrary power like to the nth power. Its equal to matrix.
Where the entries on the main diagonal get raised to the nth power and thats really convenient because if you dont have a diagonal matrix. Its really really hard to raise a matrix to like the 100th power for example. But you had if you have a diagonal matrix.
Its easy because you can just distribute the power to the entries on the main diagonal and you can convince yourself of this if you want it to so but in this video. Were gonna do this process called diagonalizing a matrix. So if a matrix is diagonalizable okay.
Which its not its not always but in the case that it is you can write the matrix. A the matrix is a as the product of three matrices. Some matrix c times.
Some matrix d times the inverse of c. Where d is gonna be a diagonal matrix. And whats the point of this well the point of this is youre trying to take like the hundredth power of some matrix.
But that matrix isnt diagonal well you can if you can write that matrix in this form cd c. Inverse. Where d is and where d is diagonal.
Then it is easy to do a to the 100th power. Because for example. A squared is equal to what cbc and verse times cd c.
Inverse. So what does that look like and if you look this c. Inverse times c.
I mean the property of inverse is a matrix times. Its inverse or vice versa is equal to the identity matrix and so this inner. Part just simplifies to the identity matrix.
And that cd identity. Dc inverse. Is just c.
D. Squared. C.
Inverse. Right. So.
This simplifies to c. D. Squared.
C. Inverse. And you could have done the same thing for any arbitrary.
Power. So a to the n.
In general equals r. Always equals c d. To the n c.
Inverse. Because youre gonna have a big chain of cd c. Inverse and all those c.
Inverse c.s are gonna cancel because of the equals. The identity matrix so and remember d.
Were defining it as a diagonal matrix and you know how to do d to the nth power you dislike up here you can just distribute it to the entries on the main diagonal and so heres the game plan puh. What are the matrices see the in theatres. How do you get the matrices c.
And b. If you want to diagonalize a matrix well you can like look at the lexer sides. I guess on the proof of this but im just gonna tell you the matrix c.
Is defined to be give these three lines. Mean is defined to be a matrix whose columns are the eigenvectors of a whose columns are eigenvectors not necessarily thought i can vectors because youre probably gonna have infinitely many. But just bear with me yet.
Were gonna talk about this. Some more the columns are eigenvectors of a. Okay.
The matrix d. Is defined to be something. That looks kind of like this lambda.
1 lambda. 2. Dot dot dot lambda.
We have the end. So youre gonna have n. Eigen values.
And then zeros everywhere else so im just gonna put big zeros. There. So its a diagonal matrix right by definition and the entries on the main diagonal are gonna be the eigenvalues and then youre gonna have c inverse right.
But if you notice in order to diagonalize a you gotta have c d. And also c inverse. So that means.
The matrix c. Has to be invertible. Okay.
Now think about what that means the invertible matrix theorem says that if youre gonna have a matrix be invertible well. Then the columns for example have to be linearly independent. So that means you better have and linearly independent eigenvectors of a assuming a is an n by n matrix so then c would be an n by n matrix it has n columns and if the columns gonna be linearly independent and the columns are the eigenvectors you need to have and linearly independent eigen vectors of a so you can construct a c matrix.
Thats invertible and so thats gonna be the deciding factor so for example or like what that means is for a to be diagonalizable. You need to have n linearly. Independent eigenvectors okay.
So this piece of information. If you have n linearly independent eigenvectors.
If you know that to be true. Then your you can construct a suitable c matrix thats invertible and then your d matrix is gonna be a diagonal matrix where the coq where the entries along the main diagonal are the eigenvalues and youre gonna see this in the next video. When we do an example.
But they have to line up. So like lambda. 1 has to be the corresponding eigenvalue to the to whatever eigenvector you put as the first column of c.
And they have to match like that right. But i mean you know following this process. You can get c and d.
And then you can compute c. Inverse. And youll be able to diagonalize.
A and then itll be super. Easy for you to compute a to an arbitrary power by this formula down here so let me leave you with the four ways that you can tell if a matrix is dying okay so here we go the four ways to tell if an n by n matrix is diagonalizable the first one like i just mentioned you have to have n linearly independent. Eigenvectors that way you can construct your c matrix and have it be invertible and then youll be able to get your diagonalization.
A equals c. Dc universe or you could think. Well.
If you have n distinct eigenvalues. Meaning and different eigenvalues. Then youre gonna have n linearly.
Independent eigenvectors and thats by some theorem. It says that if you have you know eigenvalue. 1.
And eigenvalue. 2 of your matrix then the eigen vectors corresponding to wanted to i can values are gonna be linearly independent and so if you have n unique like distinct eigen values. Then you automatically have n linearly independent eigenvectors and then the matrix would be diagonalizable the third one says that the sum that if the sum of the geometric multiplicities is n.
Then you have a diagonalizable matrix and thats because remember geometric multiplicity means the dimension of your eigen space. So you can kind of kind of think about this like if you have n dimension worth of eigen space in total then that will tell you you have n linearly independent. But this should be a t right then you have n linearly.
Independent. Eigen vectors and then the fourth. One says.
If for each lambda. The geometric multiplicity equals. The algebraic multiplicity.
Then you have a diagonalizable matrix. This one has to do with the fact that theres a theorem that says that the sum of all the algebraic multiplicities of all your eigen values. When you count for complex eigenvalues is always gonna add up to n right or in other words.
You have youre always gonna have n eigenvalues counting from algebraic multiplicity and so if the geometric multiplicity equals. The algebraic multiplicity for each eigenvalue. Then the geometric multiplicity is add up to n.
And so then you know. By number three you have a diagonalizable matrix cool so in the next video. Were gonna put this to the test.
And were gonna you look at a matrix and determine is it diagonalizable and if so were gonna diagonalize. It by diagonalize it i mean write that matrix as the product of cdc. Okay.
Ill see you then .
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