Yes, I believe so. Do you know why they specified that c ≠ 1 and c ≠ -1?Mark53 said:Homework Statement
For c not equal to ±1, what is the dimension of the eigenspaces of A
The characteristic polynomial of A is (x-1)(x+1)(x-c)
The Attempt at a Solution
each term in the characteristic polynomial has a multiplicity of 1 so does this mean that the dimension of the eigenspaces is also 1?
because it would change the multiplicity of the polynomialMark44 said:Thread moved. Questions about eigenspaces and the like are well beyond the precalculus level.
Yes, I believe so. Do you know why they specified that c ≠ 1 and c ≠ -1?
An eigenspace is a vector space associated with a specific eigenvalue of a linear transformation. It is a subset of the original vector space that contains all of the eigenvectors corresponding to that eigenvalue.
The dimension of an eigenspace is equal to the number of linearly independent eigenvectors corresponding to the eigenvalue. It can be determined by finding the nullity of the matrix (A-λI), where A is the original matrix and λ is the eigenvalue.
Eigenspaces are important because they allow us to analyze and understand the behavior of linear transformations. They also provide a way to simplify complex systems by reducing the dimensionality of the original vector space.
Yes, an eigenspace can have a dimension of zero if there are no eigenvectors corresponding to a particular eigenvalue. This means that the linear transformation does not have any non-trivial behavior along that particular eigenspace.
The dimension of an eigenspace is equal to the multiplicity of the corresponding eigenvalue. This means that if an eigenvalue has a multiplicity of 2, the dimension of its eigenspace will also be 2.