DIAGONALISATION OF 2x2 MATRIX//STEP WISE EXPLANATION//DIAGONALISING A MATRIX//MATHSPEDIA//

[Music] well ice in this video we are going to understand the concept of diagonalization now basically what this is Tom mean you understand the definition although what why we are finding this how to find this all this will try to understand in this video and this is based on the two cross two we'll try to solve an problem also know diagnosed Asian means basically means you have to just conversion it is basically conversion okay conversion of what from what you have to convert from a given matrix suppose a is the matrix you have to convert this matrix into diagonal matrix I hope you know diagonal matrix where the diagonal elements will be present dawn diagonal elements will be 0 now the diagonally from given matrix should convert into diagonal matrix and the diagonal matrix the diagonal elements will be there okay that elements will be the eigen values of the given matrix okay that will be the eigen values try to understand this okay so basically you're converting the given matrix into diagonal matrix and then the inside of that I can matrix will country in watch will have the diagonals and that diagonal elements will indicate the eigenvalues of the respective the matrix okay so we will try to understand the working rule of how to find the diagonal matrix for 2 cross 2 so here is the question if a sub matrix ok find all eigenvalues then I even reduce of a that means a matrix a and hence find P P is nothing but the modal matrix matrix you try to understand basically this is the matrix in which eigenvectors will come we will try to understand okay now D equal to D is nothing but the diagonal matrix okay P inverse again I said P is nothing but the modal matrix inverse of that into a is the given matrix into P again so we'll try to solve this problem now thing is that first step whenever you get this type of question where you have to find the diagonal matrix the first step is that you have to find the characteristic equation now how to find the cart is a key question polynomial equation we can say basically you can go by any of the methods okay so it's basically how you go by the simpler method how to find will go by lambda square minus s1 lambda plus is 2 equal to 0 where s1 is the traces or the sum of the main diagonal elements okay so what is the sum of main diagonal elements 4 minus 1 is 3 so it is basically 3 lambda plus what is s 2 determinant of this 4 into minus 1 so it is 4 into minus 1 it is minus 4 minus 4 minus 3 into 2 it is basically 10 if you have to place it as what 10 now the next step after proving after getting the katak equation next you have to solve this equation ok so in order to solve this equation what you to do notice just to try to understand you can solve by calculator also or you can go by the normal method by splitting up the middle term so I will just do it so lambda square minus 3 lambda it's basically I can use minus 5 lambda plus 2 lambda okay plus 10 is equal to 0 from this I'll get lambda plus 2 and lambda minus 5 which is equal to 0 so from this I will get lambda values as 4 minus 2 and minus sorry plus 5 okay so I got the two values now the second step comes you so is we can consider this as a second step also now the next step is that you find the eigenvector now how to find eigen vector so to just put it in this form a minus lambda I into X is equal to zero now basically this is the we'll assume some this is basically the eigenvector we'll assume it has what X Y column matrix okay know when you assume so basically how to bring in this form and you to substitute one one at a time and trying to find out the eigen vector so let us do this part know all the things okay fine so a minus lambda ought to bring so this is a minus lambda you have to do so four minus lambda 2 3 minus 1 minus lambda into X is X Y will consider this is the just assumption okay so you have to consider this s what I can vector later on we'll find out the x and y values and after doing this part you know to substitute one at each time that means in place of lambda or 2 cube minus 2 and you'll find the eigenvector then you have to put plus Phi then your Phi the other eigenvector here as I said are you have to substitute the lambda value as 2 and you have to find what the matrix so I just found out the matrix basically all you have to do is what four minus lambda that thing I have just converted I have to just substitute and find the value now from this what we can do is just simple calculation now from this I can form the equation so basically I have to multiply in from the equation so first equation will be 6x plus 2y is equal to zero okay then 3x plus y is equal to zero so these are the two equation when you see this two equation you can see that both are the same equations means if you divide this Y or take the common factor - you will get the same equation as 3x + y so this to our equations are same so it doesn't matter so basically what we can do so I'll tell you one easy technique so basically you can go by the converting this matrix into a current form and then find the equation okay that also is possible so from this how to find the eigenvector simple method I'll tell you so 3x plus y is equal to 0 you have from this I can get 3x equal to minus y now all all you have to do is that you have to get the coefficient of x and y is unity unity means what coefficient is 1 that's it now from for this you can just divide throughout by what 3 in order to get s1 okay so I will just divide so I'll get 3 3 cancel 1 divided this by whole thing by what 3 understand so - small Y by 3 now here unity means positive 1 okay I don't want negative 1 so you just take this denominator so you are just converted this into what unity coefficient now all the thing whatever is below 1 X and water is below why that becomes the eigenvector so we have X Co X Y as the eigenvector so basically whatever is below X becomes X below Y becomes y that's it this is eigenvector okay so this is a simple technique you can use to find out the eigen vector now let us subserve the second part that is lambda is equal to 5 now we just substitute here lambda basically this we had initial one so basically here we have to substitute first we substituted -2 so we got this values and next you have to substitute 5 from this year to get the eigen vector now basically we will form the equation minus x + 2 y is equal to 0 3x minus 6y is equal to 0 okay now here also you can observe that both the equations are same okay now choose any of this so let us try to understand this one or I will take this part also okay minus X plus 2y so 2y is equal to what X I will get now what I have to make I have to make the coefficient of both of this as 1 here it is not 1 but here it is 1 so I have to just divide throughout by what - okay simple thing so you have to just divide by 2 so Y by 2 is equal to X by sorry y by 2 into 2 is basically 1 okay - 2 get cancelled X by what - now whatever is their denominator of the respective x and y values becomes the x denominator is 2 y is 1 so this is the eigenvector simple okay so we got this as eigen vector and this as eigen vector after finding out the eigen vector you have to find out the next step is that you have to form the model matrix P that is basically the combination of this two you have to write as it is 1 - 3 2 1 who think comes here the doubt comes here is that see which one your to write first this one or this one you can choose choose any of this at last what happens is that when you find out the diagonal matrix ok so when you get the eigen values in this form it will be the reverse of that anyways it doesn't matter I will just show you after this ok so basically we found out what the P matrix now what you have to do you have to bring it in what D is equal to P inverse a P okay P inverse ap now we have a we have P but we don't have P inverse you have to find first P inverse now how to find P inverse that is basically adjoint a by what determinant a this is a normal method you can go for ok now the thing is that if you want to try out with the calculator also I will just show you how to find out that also so I will just show you in this calculator I of this calculator I don't know about the other calculator so I just show you how to solve in this calculator so basically we have this matrix and you have to find them inverse of this so you have to go to the mode ok then here we have metrics option ok so press on 6 so you'll get matrix a matrix B matrix C choose one of this I'll choose matrix a and you have to choose the order of the matrix that means here we have square matrix 2 cross 2 so you have to choose 2 cross 2 that is 5 and you have to give the values as per this 1 2 minus 3 and 1 ok so we have entered all the values now press on button that will be stored that values will be stored in the calculator itself ok then press shift and here you can see matrix option here for press shift 4 you will get this type of this thing now here you have to choose what 3 that means matrix a ok then you have to press inverse how to find the inverse you have to find the inverse so you have to choose this one here you can see X power minus 1 so that is basically load press now it is matrix a inverse you have to press equal to you will get it in point wise if you want infraction yes you see here it is 1/7 so how to write this values 1 by 7 minus 2 by 7 1 by 7 and 3 by 7 just you have to write this values ok this is how you can calculate it by the using by using calculator also ok so let us try so by the normal method also you get so basically I joined a means basically for the to cross 2 it is easy to find out determined a what is the term and a of this 1 into 1 minus this thing it is 1 by sir or 7 okay so I can just write 1 by 7 and adjoined a means just your intercept it will remain the same interesting the diagonal elements and change the sign of the non dying elements this will be three and this will be minus two that's it no as you have gotten calculators on one by seven and our don't get confused it's basically I have taken here determined day outside when you take one by seven outside you will get the same thing okay it's fine okay no the thing is that we have found out the P inverse notice you have to find the diagonals nemetrix that is d is equal to P inverse so I will just write P inverse what is that so 1 by 7 you take outside 1/3 minus 2 1 what is a is basically the question given 4 3 minus 1 and 2 then P is this 1 minus 3 2 1 okay now you have to multiply all this and you cater and so the thing is that you can go do by the calculator also ok so I will just do it and just to let you know the answer so after finding out all this I will get the answer as 1 by 7 multiply this and this then whatever answer you get multiply with this you will get minus 14 0 0 35 this is basically the nor diagonal matrix and just take minus 7 inside you will get what minus 2 0 0 & 5 now here you can observe what thing you can observe that earlier we found out the in the initial stage this is basically what lambda is equal to minus 2 plus 5 this is basically eigen value here you can also the same thing minus 2 and Phi are on the diagonal as I said before that diagonalization is the process in which you will get or diagonal matrix in which the diagonal elements will be the eigen values which we found out earlier this is basically the eigen values now observe here as I said if you change this column matrix here and bring it here all the change comes here is that you will get 5 here and minus 2 here that's it okay it's the same thing no need to worry you can choose this one also in front or that means first column and you can choose this one also anything is fine the basic thing is that you should get the diagonal matrix in which there I can value should be there on the diagonals limits okay so this is a procedure you have to find what the diagonal diagnostician basically notify so basically we have found out as for the questions I give values we have found out first step so I will just breathe out the working rule the first thing casick equation you ought to find out anyways it's fine if you choose any of the alphabets okay no can see equation you can denote it by Delta Lambda because I have used lambda here okay now first step first step carrotastic equation after that from that you will get eigen values just simplify it or you can use calculator also from that you will get either in values after getting eigen values you are to write it in this form in order to get what again vector so you will get this write it in this form this is nothing but the eigen vector x and y you can choose x1 y1 it depends on you okay whatever you can choose now here you have to put lambda values one by one that means you have to choose - through first or you can choose any of this first choose one value mine is to substitute here you will take toward eigen vector for the first then you have to go further so basically your to go further okay so here I just show you the how to basically simplify this you can go for the a clone form convert this equation the clone form so I don't want to do it that way so basically I have found out the other technique to do this method or they to get the eigen vector so after editing eigen vector all the after editing here you will get one vector here you will get one vector okay from lambda equal to minus 2 and lambda equal to file unit one one vector each after getting one one mentor easier to proceed with the model matrix okay after finding out the model matrix all the things what you have to do you have to just find out D which is equal to P in a P okay P inverse basically means adjoined a by determined a you can use a calculator also I showed you how to do then you have to find D substitute all this new data and that's it okay [Music] you

Channel: Mathspedia

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Keywords: cbse, maths, icse, state board, diagonalisation, diagonalisation bcpst, diagonalisation cours, diagonalisation matrice, diagonalization, diagonalisation exercice, diagonalisation matrices, diagonalisation of matrices, introduction diagonalisation, diagonalization of matrices, mathspedia

Id: sEHEU12X-AE

Length: 17min 35sec (1055 seconds)

Published: Tue May 26 2020