Let A=I_2-2 M M^T, where M is a real matrix of order 2 1 such that the relation M^T M=I_1 …

Question

Let $A=I_2-2 M M^T$, where $M$ is a real matrix of order $2 	imes 1$ such that the relation $M^T M=I_1$ holds. If $\lambda$ is a real number such that the relation $A X=\lambda X$ holds for some non-zero real matrix $X$

Accepted Answer

$\begin{gathered} A={I}_{2}-2{M}^{T} \ {A}^{2}=({I}_{2}-2M{M}^{T})({I}_{2}-2{M}^{T}) \ ={I}_{2}-2{M}^{T}-2M{M}^{T}+4{M}^{T}{M}^{T} \ ={I}_{2}-4M{M}^{T}+4M{M}^{T} \ ={I}_{2} \end{gathered}$

$\begin{gathered} AX=\lambda X \ {A}^{2}X=\lambda AX \ X=\lambda (\lambda X) \ X={\lambda }^{2}X \ X({\lambda }^{2}-1)=0 \ {\lambda }^{2}=1 \ \lambda =\pm 1 \end{gathered}$

Sum of square of all possible values $=2$

Solutions · 1