Jordan blocks and the bases

Question

Find the Jordan canonical form and a Jordan basis for the given matrix : $ \left[{\begin{array}{l} {{4}\hspace{0.33em}{0}\hspace{0.33em}{0}}\\ {{2}\hspace{0.33em}{1}\hspace{0.33em}{3}}\\ {{5}\hspace{0.33em}{0}\hspace{0.33em}{4}} \end{array}}\right] $

Answer 1

The Jordan Canonical Form is given by $\begin{pmatrix} 4&&1&&0\\0&&4&&0\\0&&0&&1\end{pmatrix}$. This is obtained by observing that the characteristic polynimial is $(x-4)^2(x-1)$ and $4$ has a geometric multiplicity of $1$. A Jordan Basis is obtained by the method of generalized eigenvectors, or, by first plugging in eigenvectors for 4 and 1 and then picking a linearly independent so as to complete the basis by solving the matrix equation for one of the vectors, as done here. Yet another method is to solve the matrix equation $AM=MJ$, where $A$ is the given matrix, $J$ is the Jordan normal form given above, and $M$ is the matrix $\begin{pmatrix} 0&&a&&0\\1&&b&&0\\1&&c&&1\end{pmatrix}$; where $a,b,c$ are to be solved and the first and last columns are the eigenvectors corresponding to $4$ and $1$ respectively. The result of doing the last procedure is $\displaystyle{\left[(0,1,1),(0.2,0,0.2),(0,0,1)\right]}$