The shuffle algorithm and Jordan blocks (Q751759)

If A is an \(n\times n\) matrix and I is the \(n\times n\) identity matrix, then elementary row operations of the matrix \([A,I]\) lead to a matrix \(M_ 1= \left[ \begin{matrix} A_ 1, & B_ 1 \\ 0, & C_ 1 \end{matrix} \right]\) where \(A_ 1\) has a full row rank. Analogously, the application of above described process to the matrix \(\left[ \begin{matrix} A_ 1, & B_ 1 \\ C_ 1, & 0 \end{matrix} \right]\) leads to \(\left[ \begin{matrix} A_ 2, & B_ 2 \\ 0 & C_ 2 \end{matrix} \right]\) where \(A_ 2\) has a maximal row rank etc. Using this process which is presented as ``shuffle algorithm'', a sequence of row matrices \(C_ 1,C_ 2,..\). is constructed. It is proved, that the space formed by all rows of matrices \(C_ 1,C_ 2,...,C_ m\) forms a basis for the space \(\{v|\) \(vA^ m=0\}\). The shuffle algorithm terminates after at most n steps. The application of this procedure on the matrix \(A-\lambda I,\) where \(\lambda\in \sigma (A)\) leads to Jordan canonical form, especially the transformation matrix can be constructed by this way.