Eigenvalues and Eigenvectors 2x2 Matrix

hey everyone its Brian in this video I'm gonna show you how to find the eigenvalues and the eigenvectors of a 2x2 matrix and if this video helps you out make sure to subscribe so here I have a 2x2 matrix 2 rows and 2 columns and I want to find all of its eigenvalues and all of its eigenvectors now in general the way you find the eigenvalues is you take the determinant of your matrix if you call your matrix a minus lambda times the identity matrix so what I'm gonna do is I'm in a sense going to subtract lambda off of this main diagonal it looks like this so if you just remember to subtract lambda off of the main diagonal and then take the determinant of this matrix and then what we're going to do is we're going to set it equal to 0 so if I take the determinant of this matrix remember the determinant of a 2x2 I multiply the elements along the first diagonal and then I subtract multiplying the elements on the other diagonal so this looks like 1 minus lambda times 4 minus lambda minus 2 times minus 1 minus 2 and I'm setting this equal to 0 now if I simplify this out if I foil this I'm gonna get lambda times lambda both negative it's gonna be lambda squared I'm going to have minus 4 lambda minus another lambda that's gonna be minus 5 times lambda and then I'm going to have 1 times 4 is 4 minus minus 2 4 plus 2 is 6 equals 0 and so this is just a second-degree polynomial which you can solve so I think this factors pretty nicely alright two numbers that add to 6 I'm sorry add 2 minus 5 and multiply to 6 thick minus 2 and -3 do it so it looks like lambda equals 2 and lambda equals 3 solves this equation so sometimes called the characteristic equation and these are the eigenvalues of this matrix so that's all you have to do just subtract lambda off the main diagonal take the determinant set it equal to 0 and solve now if I want to find the eigenvectors associated with these eigenvalues I'm gonna plug these two back into this matrix and set it equal to 0 so I'm going to start by substituting my first eigen value back into this matrix and what I'm going to do is I'm going to set this equal to 0 0 so if I reduce this if I just clean this up 1 minus 2 is minus 1 4 minus 2 is 2 and you'll notice that these rows are scalar multiples of each other so if you were to reduce this in reduced row echelon form you would see that you'd get a row of all zeros and that's going to happen with this well what does this mean if I write this in equation form I'm getting that minus X minus y is equal to 0 and also 2x plus 2y is equal to 0 and you can use either of these equations but you're gonna get the same thing that minus X is always equal to Y well what kind of numbers solve that well numbers solve that look like 1 and minus 1 right minus 1 is equal to well minus minus 1 yeah it's a little weird to think about so the eigenvector that's associated with 2 is the vector 1-1 because I can plug these values into either of these equations and it solves them let's solve for the eigen vector associated with the eigen value of three so that's the exact same process I'm just gonna plug my lambda back into this matrix and set it equal to zero now if I reduce this it's going to be the matrix minus 2 minus 1 and then 2 1 and again you noticing that these rows are just scalar multiples of each other right the second row is just negative the first row and if you write this in equation form this would be minus 2x minus y equals 0 or just as equivalent to X plus y is equal to 0 and what does that mean well this means that 2x rather minus 2x is always going to equal Y so what kind of numbers solve those well I think if I let X be 1 and Y be negative 2 that would work so some people ask me how how am i how am I getting those numbers so quickly the easiest way to do it is just to swap the coefficients here so if you see if I just write this one in the X component and the minus 2 in the Y component that's how to quickly get these eigen vectors when you reduce these equations and so this vector here one negative two is the eigenvector associated with the eigen value of 3 so these are all of the eigenvalues and eigenvectors of this matrix I hope you got something out of this video if you need to rewatch it again please do so don't forget to Like and subscribe have a fantastic day

Channel: BriTheMathGuy

Views: 23,805

Rating: 4.8949342 out of 5

Keywords: math, math tutoring, fun math, brithemathguy, online math help, maths, bri the math guy, eigenvalues and eigenvectors 2x2, eigenvalues, eigen, value, vector, matrix, 2x2, determinant, matricies, find eigen, linear algebra, algebra, system of equations

Id: pMjbWgN97pw

Length: 6min 25sec (385 seconds)

Published: Tue Apr 24 2018