Let $A$ be an $n \times n$ matrix and $v$ be an eigen vector of $A$ corresponding to eigen value $\lambda$. I would like know the following geometrically (intuition). If we substract $\lambda$ from all the diagonal entries then the resulting matrix $B$ is non singular. Can some one explain this process geometrically. I understand it mathematically why rows/columns of $B$ are linearly dependent.
you can interpret $Ax = \lambda x$ geometrically. that says that if a vector and its transform are collinear, then it is an eigenvector. this is only true for a real eigenvalue. that the condition about the determinant of $A - \lambda I$ is zero is a consequence of this. of course, you can take the geometric interpretation of the determinant as the volume and connect to $(A - \lambda I) x = 0$