We are defining a $\lambda$-Jordan segment of length $k$ is a block diagonal matrix consisting of $k$ lambda Jordan blocks of various sizes. Symbolically we write $J(\lambda;p_1,\dots,p_k)=\text{ Block Diagonal }[J_{p_1}(\lambda),\dots,J_{p_k}(\lambda)]\in M_{n}(\mathbb{C})$ where $n=\sum_{i=1}^{k}p_i$ and $p_1\ge p_2\ge\dots \ge p_k$
How can I find eigen vectors and geometric multiplicity of $\lambda$?
I have taken an example $J(5;3,2,2)$ and saw that $e_1$ is an eigen vector, but how can I find others, and is there any formulae or pattern for any $e_i$ to be an eigen vectors for such matrix?
For $J(\lambda;3,2,2)$, the eigenvectors are $e_1,e_4,e_6$. In general, each Jordan block contributes one eigenvalue that is a basis vector $e_i$ where $i$ corresponds to the index of the opening row/column of that block in $J(\lambda;p_1,\dots,p_k)$. The geometric multiplicity of $\lambda$ will be $k$.
To prove the above facts, just write down $J(\lambda;p_1,\dots,p_k) - \lambda I$ explicitly. By starting at it long enough, you can find the rank and the kernel of this matrix giving you the eigenvalues and the geometric multiplicity.