Question

Find the eigenspaces of the matrix and describe them geometrically [15 6 -4; -40 -16 11; 0 0 -1] where λ=-1 and 0

Answer 1

I got the eigenvector for -1 with (-3/8 1 0)

Answer 2

Since it is a one dimensional eigenspace (its only 1 eigenvector), the eigenspace is a line through the origin in the direction of that vector

Answer 3

er, strike that part about "only one eigenvector", i meant to to say because the eigenspace is only spanned by one vector

Answer 4

yeah λ=0 also has one it's (-2/5 1 0) I think but the answer for the eigenspace is t(2, -5,0) and t( 3, -8 0) I don't know how they got that?

Answer 5

How does that relate to the eigenvector because it's like the numerator and denominator of the fractions

Answer 6

i guess they dont like fractions in their eigenvectors, so they multiply by the denominators to make all the numbers integers. Any multiple of an eigenvector is also an eigenvector because if x is an eigenvector and c is a constant:
\[A(cx) = cAx = c\lambda x = \lambda (cx)\]

Answer 7

But whats the difference between eigenvector and eigenspace are they one of the same?

Answer 8

Because the 1 is not included in the eigenspace

Answer 9

OH I get what you mean

Answer 10

it seems like both of your eigenspaces are one dimensional, so they are both lines through the origin going in the direction of the vectors that created them.

Answer 11

Oh I see so if they weren't one dimensional the eigenvectors and eigenspaces won't necessarily be the same

Answer 12

yeah, lets say you had gotten 2 free variables and had a two dimensional eigenspace, then the geometric interpretation of that would be the plane spanned by the two vectors.

Answer 13

then, as long as a vector is in that plane (as long as the vector is a linear combination of the 2 vectors that make the plane), it is an eigenvector.

Answer 14

Alright make sense thanks a lot for your help. You sure know your stuff about eigenvectors etc.

Answer 15

No prob :) Linear Algebra is my favorite subject, i wish more people would ask questions about it lol