Skip to main content

Section 7.3 Algebraic and Geometric multiplicity

Throughout this section we assume that \(V\) is a finite-dimensional vector space over a field \(F\) and \(T\colon V\to V\) is an \(F\)-linear map.
Definition 7.3.1. (Algebraic multiplicity).
Let \(\chi_T\) be the characteristic polynomial of \(T\) and let \(\lambda\in F\) be an eigenvalue of \(T\text{.}\) Then, by the repeated application of Lemma A.1.6, we can write \(\chi_T=(t-\lambda)^m\cdot q(t)\in F[t]\) with \(q(\lambda)\neq 0\text{.}\) The natural number \(m\) is said to be the algebraic multiplicity of \(\lambda\).
Definition 7.3.2. (Geometric multiplicity).
Let \(\lambda\in F\) be an eigenvalue of \(T\text{.}\) The geometric multiplicity of \(\lambda\) is the dimension of \(V_\lambda=\ker(T-\lambda\unit_V)\) over \(F\text{.}\)

We have the following lemma.

Suppose that \(\mathfrak{B}_\lambda=\{v_1,v_2,\ldots,v_r\}\) be a basis of \(V_\lambda=\ker(T-\lambda\unit_V)\text{.}\) Thus geometric multiplicity of \(\lambda\) is \(r\text{.}\) We extend \(\mathfrak{B}_\lambda\) to a basis of \(V\text{,}\) say \(\mathfrak{B}=\{v_1,v_2,\ldots,v_r,w_1,\ldots,w_s\}\text{.}\) The matrix representation of \(T\) with respect to \(\mathfrak{B}\) has the following block form.
\begin{equation*} \begin{pmatrix}\lambda\cdot I_r\amp B\\\mathbf{O}\amp D\end{pmatrix} \end{equation*}
By a property of the determinant Section A.2, \(\chi_T=(t-\lambda)^r\cdot\chi_D\text{.}\) Therefore, the algebraic multiplicity of \(\lambda\) is at least \(r\text{.}\) Thus the result is proved.