a) Let $A$ is of nilpotent of degree $k$ then according to the condition ${A}^{k}={B}^{2k}=0$ But I can't say anything about existence of such $B$
I have no idea on $b$, $c$, please help.
a) Let $A$ is of nilpotent of degree $k$ then according to the condition ${A}^{k}={B}^{2k}=0$ But I can't say anything about existence of such $B$
I have no idea on $b$, $c$, please help.
(a) Can you show that if $\,A\,$ is a $\,n\times n\,$ nilpotent matrix , say$\,A^k=0\,$, then $\,k\leq n\,$? This answers this section as $\,B\,$ is nilpotent, but then $\,B^2=0\,$ , so if $\,A\neq 0\,$ then the claim isn't true.
(b) Since $\,A\,$ is symmetric positive definite his eigenvalues are positive real, so we can write (wrt some basis of eigenvectors) $A=\begin{pmatrix}a_1&0\\....&....\\0&a_n\end{pmatrix}\,\,,\,a_i>0$ Well, finding $\,B\,$ now is easy...