I am trying to find the Jordan normal form of a general matrix form, which is an $N \times N$ real upper bidiagonal matrix with non-zero diagonal entries and rows that sum to one:
$$\mathbf{M} = \begin{bmatrix} \lambda_1 & 1-\lambda_1 & 0 & \cdots & 0 & 0 \\[6pt] 0 & \lambda_2 & 1-\lambda_2 & \cdots & 0 & 0 \\[6pt] 0 & 0 & \lambda_3 & \cdots & 0 & 0 \\[6pt] \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\[6pt] 0 & 0 & 0 & \cdots & \lambda_{N-1} & 1-\lambda_{N-1} \\[6pt] 0 & 0 & 0 & \cdots & 0 & 1 \\[6pt] \end{bmatrix}.$$
Since this is an upper triangle matrix, the eigenvalues are the diagonal entries $\lambda_1,...,\lambda_{N}$ with $\lambda_N=1$, and so the eigenvalue matrix is the diagonal matrix with the same diagonal entries as the initial matrix. The corresponding (un-normalised) eigenvalue matrix can be shown to be:
$$\mathbf{v} = \begin{bmatrix} v_{1,1} & v_{1,2} & v_{1,3} & \cdots & v_{1,N-1} & v_{1,N} \\[6pt] 0 & v_{2,2} & v_{2,3} & \cdots & v_{2,N-1} & v_{2,N} \\[6pt] 0 & 0 & v_{3,3} & \cdots & v_{3,N-1} & v_{3,N} \\[6pt] \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\[6pt] 0 & 0 & 0 & \cdots & v_{N-1,N-1} & v_{N-1,N} \\[6pt] 0 & 0 & 0 & \cdots & 0 & v_{N,N} \\[6pt] \end{bmatrix}.$$
where $v_{i,k} = \prod_{\ell=0}^{i-1} (\lambda_k - \lambda_\ell) \times \prod_{\ell=0}^{i-1} (1 - \lambda_\ell)$. If the eigenvalues $\lambda_1,...,\lambda_{N}$ are all distinct then the eigenvectors are linearly independent, and so the matrix $\mathbf{M}$ is diagonalisable. However, I am interested in the case where the eigenvalues are not necessarily distinct, and so I am trying to get an expression for the generalised eigenvector matrix in the Jordan normal form. So far, I have figured out that the geometric multiplicity of each distinct eigenvalue is one, so each distinct eigenvalue has a single Jordan block. However, I have not succeeded in obtaining the generalised eigenvalue matrix in the Jordan normal form. I have looked for literature on this problem but I have not found anything that deals directly with this case.
Questions: Is there an explicit form for the values in the generalised eigenvalue matrix? Is there any mathematical literature on the Jordan normal form of bidiagonal matrices?
Note: If it helps, in the application I am dealing with I have $0 \leqslant \lambda_i \leqslant 1$ for all $i=1,...,N-1$. I am also willing to assume that $0 < \lambda_i < 1$ if needed, but I would like a general form that makes no further assumptions than this.