I would like to prove that every $n\times n$ matrix is the sum of a diagonalizable matrix and a nilpotent matrix. How is this possible? I'm not sure where to begin really- I know that a nilpotent matrix is one of which some power is the zero matrix.
I also know that a matrix A can be written as $AP=PJ$ with $P$ invertible and $J$ of Jordan form.
I have proven that any strictly upper triangular matrix is nilpotent, so $J$ can be written as $D+N $, with D diagonal and $N$ nilpotent, but how can I change this for A?
Thank you!
Linear Algebra – Every n×n Matrix as Sum of Diagonalizable and Nilpotent Matrices
linear algebramatricesmatrix decomposition
Best Answer
You have $A = PJP^{-1}$ where $J$ is in Jordan form. Write $J = D + N$ where $D$ is the diagonal and $N$ is the rest, which is strictly upper triangular and thus nilpotent. Then $A = PDP^{-1} + PNP^{-1}$. The former is clearly diagonalizable, while the latter is nilpotent; just note that $(PNP^{-1})(PNP^{-1}) = PN(P^{-1}P)NP^{-1} = PN^2P^{-1}$ and so on.