91Ó°ÊÓ

Eigenvector:An eigenvector ofA is a nonzero vector vin${\mathbf{R}}_{\mathbf{n}}$such that$\mathit{A}\mathit{v}\mathbf{=}\mathit{λ}\mathit{v}$, for some scalar$\mathit{λ}$.

Suppose that$\stackrel{→}{v}$is an$\mathrm{n}×\mathrm{n}$eigenvector of a matrix A with associated Eigen-value $\mathrm{λ}$.

To determine what can be said about $\ker \left(\mathrm{A}-{\mathrm{λ\pm õ}}_{\mathrm{n}}\right)$.

Also determine if the matrix $A-\mathrm{λ}{I}_{\mathrm{n}}$is invertible.

By definition of Eigen-vector,

Consider a non-zero vector$\stackrel{→}{v}$in$R_n$is Eigen-vector of A if $A\stackrel{→}{v}=\mathrm{λ}\stackrel{→}{v}$.

For some scalar $\mathrm{λ}$, where$\mathrm{λ}$is the Eigen-value with respect to Eigen-vector $\stackrel{→}{v}$.

The expression$\left(A-\mathrm{λ}{l}_n\right)\stackrel{→}{v}$is:

$$\begin{gathered} \left(A-\mathrm{λ}{l}_n\right)\stackrel{→}{v}=A\stackrel{→}{v}-\mathrm{λ}{l}_n\stackrel{→}{v} \\ \left(A-\mathrm{λ}{l}_n\right)\stackrel{→}{v}=A\stackrel{→}{v}-\mathrm{λ}\stackrel{→}{v} \\ \left(A-\mathrm{λ}{l}_n\right)\stackrel{→}{v}=\stackrel{→}{v}-\mathrm{λ}\stackrel{→}{v} \\ \left(A-\mathrm{λ}{l}_n\right)\stackrel{→}{v}=0 \end{gathered}$$

The value obtained here is $\left(A-\mathrm{λ}{l}_n\right)\stackrel{→}{v}=0$.

Thus, a vectorthat is non-zero here is present here, which implies$\stackrel{→}{v}$is in the kernel of $A-\mathrm{λ}{l}_n$.

By definition, a set is said to be not - trivial if any vector present in it is non-zero.

Thus, $\ker \left(\mathrm{A}-{\mathrm{λ\pm ô}}_{\mathrm{n}}\right)$will not be a set that is trivial.

Hence,$\ker \left(\mathrm{A}-{\mathrm{λ\pm ô}}_{\mathrm{n}}\right)≠\left\{\stackrel{→}{0}\right\}$ As a deduction, the $A-\mathrm{λ}{l}_n$matrix will not be invertible.