91Ó°ÊÓ

Eigenvector:An eigenvector ofAis a nonzero vector vin${\mathbf{R}}_{\mathbf{n}}$such that$\mathbf{Av}\mathbf{=}\mathbf{λ\pm ¹}$, for some scalar$\mathbf{λ}$.

By definition of Eigen-vector,

Consider a non-zero vector $\stackrel{â‡\P Ä}{v}$in $R^n$is Eigen-vector of A if $\stackrel{â‡\P Ä}{A}v=\mathrm{λ}\stackrel{â‡\P Ä}{v}$

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

It is required to find 4x4 matrices for the condition that $\stackrel{â‡\P Ä}{e_2}$is an Eigen-vector. From the definition stated above, it is known that, if the Eigen-vector is $\stackrel{â‡\P Ä}{v}$for a matrix A, then,

$A\stackrel{â‡\P Ä}{v}=\mathrm{λ}\stackrel{â‡\P Ä}{v}$

If ${\stackrel{â‡\P Ä}{e}}_2$is an Eigen-vector for the 4x4 matrix A, then

$A\stackrel{â‡\P Ä}{e_2}=\mathrm{λ}\stackrel{â‡\P Ä}{e_2}$

It is known that for a 4x4 matrix, there are four columns present. The column vectors are represented as $\stackrel{â‡\P Ä}{v_1,}\stackrel{â‡\P Ä}{v_2,}\stackrel{â‡\P Ä}{v_3,}\stackrel{â‡\P Ä}{v_4,}$denoting first, second, third and fourth column respectively.

Here as the Eigen vector is represented wth notation $\stackrel{â‡\P Ä}{e}$, then $\stackrel{â‡\P Ä}{e_2}$here is said to represent the second column of the matrix.

Thus, for a 4x4 matrix A, it can be said that the second column has to be of the form shown below:

This shows that only the second-row element has a value that is the Eigen-value which is .

$\left[\begin{array}{c}0\\ \mathrm{λ}\\ 0\\ 0\end{array}\right]$

Thus, as no other constraints are there on the other elements in other columns, they can be any value. Here for an example, they are shown represented using alphabets.

Thus, one of the forms of the matrixis shown below:

$\left[\begin{array}{cccc}a& 0& c& d\\ e& \mathrm{λ}& f& g\\ h& 0& i& j\\ k& 0& l& m\end{array}\right]$

Thus, the required result is found.