Eigenspace

If \(v\) is an eigenvector for \(\lambda\), any nonzero multiple of \(v\) is too, and so is the sum of two eigenvectors for the same \(\lambda\) (as long as it is not zero). So it is natural to collect them in one set together with zero: the result is a subspace, the kernel of \(A-\lambda I\). Zero is added by hand because it is never an eigenvector, yet a subspace must contain it.

Computing an eigenspace means solving a homogeneous system. For \(A=\begin{pmatrix}4&1\\2&3\end{pmatrix}\) and \(\lambda=5\), the system \((A-5I)v=0\) reduces to \(-x+y=0\), and the eigenspace is the line spanned by \((1,1)\). Its dimension, called the geometric multiplicity, is \(n-\operatorname{rank}(A-\lambda I)\) and satisfies \(1\le \dim E_\lambda\le\) the multiplicity of \(\lambda\) as a root of the characteristic polynomial.

A typical mistake is adding eigenvectors of different eigenvalues and expecting another eigenvector: it does not work, because the sum lies in neither space. What is true is that eigenspaces of distinct eigenvalues meet only at zero and that eigenvectors of distinct eigenvalues are independent. That independence, together with the dimensions of the eigenspaces, will decide in the next concept whether a matrix can be diagonalized. Put differently, \(E_\lambda\) is the kernel of \(A-\lambda I\): the set of solutions of \((A-\lambda I)x=0\), where \(I\) is the identity of the same size as \(A\).