Subspace property of an eigenspace Given an \((n,n)\)-matrix \(\Mtrx{A}\), the eigenspace \(E_k\) of \(\Mtrx{A}\) with eigenvalue \(\lambda_k\) is a subvector space of \(\RNrSpc{n}\). This is so because \(E_k\) is the null space of the matrix \((\Mtrx{A} - \IdMtrx{n})\), hence is the orthogonal complement of the row vectors of \((\Mtrx{A} - \IdMtrx{n})\), and this is a subspace of \(\RNrSpc{n}\).
If \(\Mtrx{B}\) is an arbitrary \((n,n)\)-matrix, we know that
| \(\Dim{\NullSpc{B}}\) | \(=\) | \(n - \Rnk{B}\) |
Applied to the matrix \(\Mtrx{B} \DefEq (A - \lambda\IdMtrx{n})\) and the geometric multiplicity \(\GmtrcMltplcty{\lambda}\) of the eigenvalue \(\lambda\), this yields
\[\GmtrcMltplcty{\lambda} = \Dim{\NullSpc{A-\lambda\IdMtrx{n}}} = n - \Rnk{A-\lambda\IdMtrx{n}}\]Algebraic vs. geometric multiplicity One can show that the algebraic and geometric multiplicity of an eigenvalue are related by the inequality
\[1 \leq \text{geometric multiplicity} \leq \text{algebraic multiplicity}\]