Many proofs that the geometric multiplicity of an eigenvalue cannot exceed the algebraic multiplicity of an eigenvalue use the following idea:
Let the geometric multiplicity of the eigenvalue $\lambda$ of $A$ be $k$. Then we have $k$ linearly independent vectors $v_1,...,v_k$ such that $Av_i = \lambda v_i$. If we change our basis so that the first $k$ elements of the basis are $v_1,...,v_k$, then with respect to this basis we have $$ \begin{bmatrix} \lambda I_k & * \\ 0 & B \\ \end{bmatrix} $$
(this came directly from this proof)
Could someone explain why this is the case?
