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@Magma

formula is $A-\lambda I=0$

right?

So, equation will be

$\begin{bmatrix} 1 &0 \\ 0 &1 \end{bmatrix}$ $\begin{bmatrix} X_{1}\\ X_{2} \end{bmatrix}$=1.$\begin{bmatrix} X_{1}\\ X_{2} \end{bmatrix}$

Now we need to find $X_{1}$, $X_{2}$

right?
edited

@srestha Ma'am

I don't understand what you write,and it look like wrong, but i can explain

see this it might be useful

https://gateoverflow.in/116260/geeksquiz-no-of-linearly-independent-vector-for-eigen-value

we don't need to find out 2 Eigen vectors,

$\lambda_1 = 1$

$\lambda_2 = 1$

Algebraic multiplicity of '1' = 2

Geometric multiplicity of '1' = 2

We know that for Eigenvector  $AX=\lambda X$------------------>$(1)$ Where A=Given Matrix and $X=\begin{bmatrix} x_{1}\\x_{2} \end{bmatrix}$ Eigen Vectors

$\Rightarrow AX-\lambda X=[0]$

$\Rightarrow (A-\lambda I) X=[0]$---------------->$(2)$Where   $I=\begin{bmatrix} 1 & 0\\ 0 &1 \end{bmatrix}$Idenitity Matrix.

For Eigen Values we write the characteristic  equation$:|A-\lambda I|=0$----------->$(3)$

Now,Given that Matrix $I=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

First we can find the Eigen values:

Here in this question $A=I$

So,we can write the equation$:|I-\lambda I|=0$------->$(4)$

now we can find $(I-\lambda I)=\begin{bmatrix} 1 & 0\\ 0 &1 \end{bmatrix}-\lambda \begin{bmatrix} 1 & 0\\ 0 &1 \end{bmatrix}$

$(I-\lambda I)=\begin{bmatrix} 1 & 0\\ 0 &1 \end{bmatrix}- \begin{bmatrix} \lambda & 0\\ 0 &\lambda \end{bmatrix}$

$(I-\lambda I)=\begin{bmatrix} 1-\lambda & 0\\ 0 &1-\lambda \end{bmatrix}$

Now from the equation $(4),$

$|I-\lambda I|=\begin{vmatrix} 1-\lambda & 0\\ 0 &1-\lambda \end{vmatrix}=0$

$\Rightarrow [(1-\lambda)(1-\lambda)-0]=0$

So$,\lambda=1,1$

Eigen values are $\lambda_{1}=1,\lambda_{2}=1$

We have Another Method,to finding the Eigen Values:

Given that Matrix $I=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

Let suppose Eigen Values are $\lambda_{1},\lambda_{2}$

Important Property of finding the Eigen Values:
$(1)$ Sum of All Eigen Values$=$Sum of All Leading Diagonal Elements(Trace of the Matrix)
$(2)$ Product of All Eigen Values $=Det(A)=|A|$

Now,$\lambda_{1}+\lambda_{2}=1+1=2$----------->$(5)$

and $\lambda_{1}.\lambda_{2}=|I|=(1-0)=1$------------->$(6)$

Now we can find the values of $\lambda_{1},\lambda_{2}$

We know that if $a,b$ are the root the quadratic equation,so we can construct the quadratic equation

$x^{2}-(a+b)x+ab=0$

If $\lambda_{1},\lambda_{2}$ are the root of the quadratic eqation we can write the quadratic equation

In our question we have the value of $\lambda_{1}+\lambda_{2}=2,\lambda_{1}.\lambda_{2}=1$

$x^{2}-(\lambda_{1}+\lambda_{2})x+\lambda_{1}.\lambda_{2}=0$

$x^{2}-2x+1=0$

$x^{2}+1^{2}-2x=0$

$(x-1)^{2}=0$

$x=1,1$

So$,\lambda_{1}=1,\lambda_{2}=1$

We have Another Method,to finding the Eigen Values:

Given that Matrix $I=\begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

In the case of Triangular matrix(Lower triangular matrix or Upper triangular matrix),Leading diagonal elements itself are Eigen Values.

We all know that Identity Matrix is the triangular matrix $($ Symmetric matrix also $A^{T}=A$$)$

So,Eigen Values are $\lambda_{1}=1,\lambda_{2}=1$

Now, from the equation $(2)$

$(A-\lambda I) X=[0]$

Here $A=I,$      $(I-\lambda I) X=[0]$

$(I-\lambda I)X=\begin{bmatrix} 1-\lambda & 0\\ 0 &1-\lambda \end{bmatrix}.\begin{bmatrix} x_{1}\\x_{2} \end{bmatrix}=\begin{bmatrix} 0\\0 \end{bmatrix}$

Here $\lambda=1$

$\Rightarrow \begin{bmatrix} 1-1 & 0\\ 0 &1-1 \end{bmatrix}.\begin{bmatrix} x_{1}\\x_{2} \end{bmatrix}=\begin{bmatrix} 0\\0 \end{bmatrix}$

$\Rightarrow \begin{bmatrix}0 & 0\\ 0 &0\end{bmatrix}.\begin{bmatrix} x_{1}\\x_{2} \end{bmatrix}=\begin{bmatrix} 0\\0 \end{bmatrix}$

Rank of the matrix $,r(I)=0$ ,Number of unknowns $=2$

Here$,r(I)<UK (0<2),$ this is the condition for Infinite many numbers of solutions, and we can assign $UK−r(I)=2−0=2$ linearly independent values to different variable  for finding the $x_{1},x_{2}$      $[$UK---->Number of Unknown$]$

Here $x_{1}=k_{1},x_{2}=k_{2}$

Now  $,0.x_{1}+0.x_{2}=0$------------>$(7)$

Now,$X=\begin{bmatrix} x_{1}\\x_{2} \end{bmatrix}$

$\Rightarrow \begin{bmatrix} x_{1}\\x_{2} \end{bmatrix}=\begin{bmatrix} k_{1}\\k_{2} \end{bmatrix} = k_1 \begin{bmatrix} 1\\ 0 \end{bmatrix} + k_2 \begin{bmatrix} 0\\ 1 \end{bmatrix}$

You can clearly see,there are two linearly independent eigen vector

@Lakshman

So the linearly independent eigenvectors are  k1$\begin{pmatrix} 1\\ 0 \end{pmatrix}$  and   k2$\begin{pmatrix} 0\\ 1 \end{pmatrix}$    for any values of k1, k2.  ?

Yes any value of $k$

thanks.

@
hope you got the answer now.

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