DIAGONALISATION OF 3x3//STEP WISE EXPLANATION//TRICKS AND TIPS//MATHSPEDIA//
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- Опубликовано: 9 сен 2024
- Diagonalization of matrices in english
Diagonalization of 3×3 matrix example.
Diagonalization of square matrices.
Matrices.
DIAGONALISATION OF 3x3
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This is what i have been looking for all semister!!!
Thank you for this video. I learned what I needed. However, there is a much easier way to solve for the eigenvectors, which is related to LU factoring a matrix. And also to obtain an orthonormal transformation matrix from normalized eigenvectors. Let me demonstrate using the matrix equation associated with lambda = 3. The matrix equation is repeated here for illustration
| 0 -1 1| |x|
|-1 2 -1| |y| = 0 Eq. (1)
| 1 -1 0| |z|
Pick the absolutely largest entry in this matrix, in this case 2, as the pivotal element. If there are two or more absolutely largest entries, pick one with the most nonzero entries in its row. The row and column containing the pivotal element are the pivot row and pivot column. Scale the pivot row respectively by the ratio of each nonzero entry in the pivot column to the pivotal element. Thus divide the row 1, column 2 entry, -1 by 2 to get -1/2 and multiply pivot row 2 by (-1/2) to get |1/2, -1, 1/2| Then subtract this row from row 1 to get |-1/2, 0, 1/2| In a similar manner divide the row 3, column 2 entry by 2 to again get -1/2, scale the pivot row, and subtract this row from row 3 to get |1/2, 0, -1/2| Form a new matrix
| -1/2 0 1/2| |x|
| -1 2 -1 | |y| = 0 Eq. (2)
| 1/2 0 -1/2| |z|
Now strike the pivot row 2 and pivot column 2 from Eq. (2) to get
| -1/2 1/2| |x| = 0 Eq. (3)
| 1/2 -1/2| |z|
Since all entries have the same absolute magnitude, pick the row 1, column 1 entry, -1/2 as the next pivotal element. Then scale pivot row 1 by (1/2)/(-1/2) and subtract from row 2 to get |0 0| and the modified matrix equation
| -1/2 1/2| |x| = 0 Eq. (4)
| 0 0 | |z|
From Eq. (4) we have z - x =0 and from the second equation in Eq. (2) we have -x + 2y -z = 0, which give
x = z and y = (x+z)/2. Since we have two equations and three unknowns, one value can be arbitrarily set, say set z = 1. Then x = z = 1 and y = (x+z)/2 = (1 + 1)/2 = 1.
To get an orthonormal transformation matrix from the three eigenvector solutions, each eigenvector should be normalized to unity. The current magnitude of this eigenvector is
sqrt(x^2 + y^2 + z^2) = sqrt (1^2 + 1^2 + 1^2) = sqrt(3) Thus the normalized eigenvector would be
| 1/sqrt(3), 1/sqrt(3), 1/sqrt(3) |^T or | sqrt(3)/3, sqrt(3)/3, sqrt(3)/3 |^T When the 3 by 3 matrix is composed of three normalized eigenvectors, it is orthonormal and its inverse is equal to its transpose, which eliminates the need to compute a complex matrix inverse.
Since the determinant of this symmetric matrix is 36 it is also positive definite. A positive definite matrix has all nonzero and positive eigenvalues. And the product of its three eigenvalues is also 36, so a good starting estimate for an eigenvalue would be the cube root of 36 or about 3.3. Synthetic division is only a good way to solve for the first eigenvalue when the values are nice, as in this example. An iterative Newton-Raphson method would be faster and easier.
Let y = x^3-s1x^2+s2x-s3 and y' = 3x^2-2s1x+s2. Then dx = -y/y' and x
Thank you so much bro ❤️❤️❤️❤️ valuable video thank you 🎉🎉🎉🎉
I get so use to Gaussian elimination to find the eigenvectors of corresponding eigenvalues (and assigning 1 for independent variable rows) it is neat seeing someone applying Cramer's sort of thing on two rows with column blocking for x/|..| = -y/|..| = z/|..| way of doing it.
learnt
eigen value, eigen vector, Synthetic Division, cramer's rule ( cross multiplication ), Diagonalization
in less than 20 mins 😭😭 dude is a life save 💫🫂
Tq sir I understood
Does this method have a name?
By the way, video helped me much .Thank you.
PtransposeA p how to find in calculator using like 1 divided by root 2
Bhaiya ke problem ho rahi hai question me jab crammerce rule lagate hai to x=y=z jab likjte hai to kya sirf y me minus aayega?????? Or xYz ke cofficient nahi lenge kya crammerce rule lagate samay?
thanks sir it really helps me a lot
Thanks sir.. for easy understanding teaching
Wonderfull explanation bro
Sir plz give vedio this problem solve by orthogonal reduction method sir
Yoy are great
Thank u sir♥
Sir how to find P inverse in calsi plz can u send the video
Sir what will we do if any diagonal elements have -ve sign then we ignore and add all diagonal elements or we add all elements with -ve sign
Please prepare video about Markov chain rule
Sure
Chracteristic equation e ses er constant duto 36 er jagay 37,38 hoga
11:39 Sir in case 1 where lambda is equal to 2 value of x should be -1 and z should be 1.
No x value is 1 and z value is 1.Make coefficient of x and z as +1.So I have taken z coefficient in the denominator (-1).
Sir cooefficent ke aage aap ne 2 kha se lia
Loveeeeeeeeeeee ❤❤
2:35 in case if we got the trace in -minus? Will the it become +11
Yes
bhi hindi my bola kry ya urdu
😃
Sir last lo aha values ravadhumledhu ga 2,3,6
How the s3value came sir
Sir lpl=-6 kada meru 6. Veasaru
|p|= 6 correct 💯
Ok
More grantiful
Sir pls e sum calsi lo cheyandhi pls
Next video sure👍
P inverse ksy nickla ap na
@@user-co8tz4sg3f determinant(P)/adjoint(P)
Hindi me bana liya karo vedio
E concept ni Casio lo chesi chupiyandi
Ok 👍 next video
I think sir u have made a mistake
When applying Cramer rule ( the answer wa -1 1 -1
In the second part when lamda is equal to 3
But u had written 1 1 1 in the p 😅
Lamda =3 ,it is [1 1 1], check 15:21
@@MATHSPEDIAabhisir that's the confusion
The value of x y z are -1,1,-1
But in a column form 1,1,1
@@PANADOL695 "y" is having (-) as coefficient,when it is taken down in the Denominator it becomes (+1) so it will be [-1 -1 -1] and you can eliminate (-) sign since all elements have (-) sign so it becomes [1 1 1].
@@MATHSPEDIAabhi sir thank u so much it was a confusion
But I get it now
Thank a lot 🥹
Bhai meri agr english nhi ati toh hindi main bol le teri awaz h samjh nhi a rahi
తెలుగేనా సార్ మీది....
What sir???