- #1
Gear300
- 1,213
- 9
I remember reading a theorem that said that for an n x n matrix A, there exists a basis of Cn consisting of generalized eigenvectors of A.
For A = [1 1 1; 0 1 0; 0 0 1] (the semicolons indicate a new row so that A should be 3 x 3 with a first row consisting of all 1's and a diagonal of 1's). The eigenvalue of A is 1 with an algebraic multiplicity of 3 and geometric multiplicity of 2. So I can pull out a basis of 2 vectors from Ker(A-uI). For the third generalized eigenvector, it should be in Ker((A-uI)3), but that produces 0v = 0, in which v could be any vector in Cn...did I do something wrong here?
For A = [1 1 1; 0 1 0; 0 0 1] (the semicolons indicate a new row so that A should be 3 x 3 with a first row consisting of all 1's and a diagonal of 1's). The eigenvalue of A is 1 with an algebraic multiplicity of 3 and geometric multiplicity of 2. So I can pull out a basis of 2 vectors from Ker(A-uI). For the third generalized eigenvector, it should be in Ker((A-uI)3), but that produces 0v = 0, in which v could be any vector in Cn...did I do something wrong here?