How can I find the dimension of the eigenspace? The matrix \[A= \b

Pamela Meyer 2021-12-19 Answered
How can I find the dimension of the eigenspace?
The matrix A=[9117] has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimension of the eigenspace.
So we found the eigenvalue by doing AλI to get:
λ=8
But how exactly do we find the dimension of the eigenspace?
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Expert Answer

sonSnubsreose6v
Answered 2021-12-20 Author has 21 answers
The dimension of the eigenspace is given by the dimension of the nullspace of A8I=(1111) , which one can row reduce to (1100), so the dimension is 1.
Note that the number of pivots in this matrix counts the rank of A−8I. Thinking of A−8I as a linear operator from R2R2, the dimension of the nullspace of A is given by dim(R2) −rank(A)=2−1=1 by the so-called rank-nullity theorem. Of course, one can be more explicit: it is straightforward to see that the nullspace of A−8I is spanned by the vector (1,1), and hence has dimension 1.
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Wendy Boykin
Answered 2021-12-21 Author has 35 answers
By definition, an eigenvector v with eigenvalue λ satisfies Av=λv, so we have Avλv=AvλIv=0, where I is the identity matrix. Thus, (AλI)v=0, and v is in the nullspace of AλI.
Since the eigenvalue in your example is λ=8, to find the eigenspace related to this eigenvalue we need to find the nullspace of A−8I, which is the matrix
[1111]
We can row-reduce it to obtain
[1100]
This corresponds to the equation
x - y = 0.
so x=y for every eigenvector associated to the eigenvalue λ=8. Therefore, if (x,y) is an eigenvector, then (x,y)=(x,x)=x(1,1), meaning that the eigenspace is W=[(1,1)], and its dimension is 1.
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