Find the eigenvalues, eigenspaces, algebraic and geometric multiplicities for the 3 × 3 identity matrix I3. Do your results make sense?

The characteristic polynomial for A = I₃

is pI₃ (x) = (1- x)³

which has eigenvalue   λ = 1 with algebraic multiplicity α_A (1) = 3. Looking for eigenvectors, we find that A − λI =

The nullspace of this matrix is all of C3, so that the eigenspace is

 and the   geometric multiplicity is γ_A(1) = 3. 

Does this make sense? Yes! Every vector x is a solution to I₃x = 1x, so every nonzero vector is an eigenvector with eigenvalue 1. Since every vector is unchanged when multiplied by I₃, it makes sense that λ = 1 is the only eigenvalue.