The dimension of the eigenspaces for the matrix A, characterized by the polynomial (x-1)(x+1)(x-c), is determined to be 1 for each eigenvalue when c is not equal to ±1. Each term in the characteristic polynomial has a multiplicity of 1, leading to a direct correlation between the multiplicity and the dimension of the eigenspaces. The specification that c ≠ 1 and c ≠ -1 is crucial as it prevents changes in the multiplicity of the polynomial, which would affect the eigenspace dimensions.
PREREQUISITESStudents of linear algebra, educators teaching matrix theory, and anyone seeking to deepen their understanding of eigenvalues and eigenspaces in mathematical contexts.
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?