🔷14 - Eigenvalues and Eigenvectors of a 2x2 Matrix

🔷14 - Eigenvalues and Eigenvectors of a 2x2 Matrix
Given that A is a square matrix (nxn),
Ax = kx -------(1), where
A = an nxn matrix (square matrix),
x = eigenvector of A corresponding to k,
k = eigenvalue of A corresponding to x
It is usually asked to find the eigenvalue as well as the eigenvector that satisfy the above equation.
Notice that we are only interested in the solution with x not equal to zero.
from (1), Ax = kx
Ax = kIx ------(2) ,
(A-kI)x = 0 ----(3)
the system will give a non-zero solution if and only if det (A-kI)x = 0 ,
det (A-kI) gives rise to a polynomial called the characteristic polynomial and the equation formed when det (A-kI) = 0 is called the characteristic equation. The solutions to the equation are the eigenvalues....
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00:00 - Eigenvalues and eigenvectors
05:44 - Example

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[@SkanCityAcademy_SirJohn]

Awww thanks so much Bonye

[@MuctaruKabba]

I was stuck in understanding how to go from the second to third step of finding eigenvectors. Your third equation and explanation have help me immensely. I'm still a work in progress, but I may have crossed the Rubicon, thanks to you.

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[@sloshycomic123]

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[@SkanCityAcademy_SirJohn]

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[@ahmadyusuf4903]

Thank you Sir, Please how do I find the eigen vectors using magnitude formula

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[@SkanCityAcademy_SirJohn]

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[@AFCOE]

Any value we choose for X2 will be valid?

[@SkanCityAcademy_SirJohn]

Yes, but usually we choose smaller numbers like 1, -1 etc for simplification and analysis.

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[@JOJOJOE-mm4zq]

WHY DID YOU DIVIDE ROW 1 BY NEGATIVE ONE OVER FOUR FOR THE SECOND EIGEN VECTOR. YET FOR THE FIRST EIGEN VALUE YOU ONLY MANIPULATED ROW 2 ONLY.

[@SkanCityAcademy_SirJohn]

It's very simple, the idea is to reduce the augmented matrix into it's row echelon form, that way because we have a 2*2 matrix, the a21 value needs to be zero. So based on the elements in Row 1 we perform elementary row operations to get a zero for a21.