I understand that if an n-dimensional matrix has n eigenvalues, each eigenvalue must correspond to one eigenvector.
When there is multiplicity in the roots of the characteristic polynomial of a matrix, the repeated eigenvalue can have as many eigenvectors as its multiplicity
Is there any limitation requiring the repeated eigenvalue in particular to take multiple eigenvectors?
In other words, can any eigenvalue take multiple eigenvectors in this situation?
Best Answer
No, only the repeated one(s). For instance, the diagonal matrix with diagonal entries $3,3,3,2,2,1$ has a $3$-dimensional eigenspace for the eigenvalue $3$, a $2$-dimensional eigenspace for the eigenvalue $2$ and a $1$-dimensional eigenspace for the eigenvalue $1$. The dimension of the eigenspace corresponding to an eigenvalue (which is called its geometric multiplicity) is at most the multiplicity of the eigenvalue as a root of the characteristic polynomial (which is called its algebraic multiplicity).