Find All the characteristic values and characteristic vectors

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Find the eigenvalues and corresponding eigenvectors of the matrix

Find All the characteristic values and characteristic vectors of the matrix

{ \color {Blue} A= \begin {bmatrix}
 1& 1 &1 \\ 
 1& 1 &1 \\ 
 1& 1 & 1
\end {bmatrix}}

Solution

the characteristic equation of the matrix is

 \left  |A - \lambda I    \right  |=0
{\color {Blue} or \; \;  \begin {vmatrix}
1 - \lambda & 1  & 1  \\ 
1 & 1 - \lambda & 1\\ 
1  &  1 & 1 - \lambda
\end {vmatrix}=0}
{\color{Blue} or \;\; \begin{vmatrix}
3 - \lambda & 1 &1 \\ 
3 - \lambda & 1 - \lambda & 1\\ 
3 - \lambda & 1 & 1 - \lambda
\end{vmatrix}=0 }\;\; {\color{Red} on \; C_1 \rightarrow C_ 1 + C_ 2 + C_ 3}
{\color {Blue} or \; \; \;  (3 - \lambda )\begin{vmatrix}
1 & 1 & 1 \\ 
1 & 1 - \lambda & 1\\ 
1 & 1 & 1 -  \lambda
\end {vmatrix}=0 }
{\color {Blue} or \; \; \;  (3 - \lambda )\begin{vmatrix}
1 & 1 & 1 \\ 
0 & - \lambda & 0 \\ 
0 & 0 &  -  \lambda
\end {vmatrix}=0 } \; \; \; {\color{Red} on \; \; R_ 2 \rightarrow R_ 2 - R_1\; , \;  R_ 3 \rightarrow R_ 3 - R_1}
or  \; \; (3-  \lambda) \lambda ^ 2 = 0 \\ \\ \therefore \lambda=0,0, \lambda=3

The characteristic values are 0, 0, 3

{\color{DarkBlue} Let \; \; \;X=\begin{bmatrix}
x_1\\ x_2
\\x_3 

\end{bmatrix} \; \;be \;  \;the  \; \;characteristic   \;vector \;corresponding  \\  \;to  \; the \; characteristic  \;value \;{\color{DarkOrange} \lambda=0 } \;then}
(A-0I)X=0
{\color {Blue} or \; \; \;  \begin{vmatrix}
1 & 1 & 1 \\ 
1 & 1 & 1 \\ 
1 & 1 &  1
\end {vmatrix}\begin{bmatrix}
x_1\\ x_2
\\x_3 

\end{bmatrix}= \begin{bmatrix}
0\\ 0
\\0

\end{bmatrix}} 
{\color {Blue} or \; \; \;  \begin{vmatrix}
1 & 1 & 1 \\ 
0 & 0 & 0 \\ 
0 & 0 &  0
\end {vmatrix}\begin{bmatrix}
x_1\\ x_2
\\x_3 

\end{bmatrix}= \begin{bmatrix}
0\\ 0
\\0

\end{bmatrix}} {\color{Red}\; \; on \; R_ 2 \rightarrow R_ 2 - R_1 \; ,\;  R_ 3 \rightarrow R_ 3 - R_1}

The coefficient  matrix is of rank 1 and equation have  3-1=2 linearly independent solutions.these equation reduced to a single equation

x_1+x_2+x_3=0
The \; solutions \; are\;  {\color{DarkBlue} x_1=1\; , \; x_2=0 \; , \; x_3=-1} \;and \; {\color{DarkBlue} x_1=0,x_2=1,x_3=-1
}
Hence \; \; X_1= \begin {bmatrix}
1 \\ 0 
\\ -1 

\end {bmatrix} \; and \; \; X_2= \begin{bmatrix}
0 \\ 1 
\\ -1 

\end {bmatrix}

Are characteristics vectors corresponding to the characteristic value 0

For \; \lambda=3 \;,we \,have 
(A-3I)X=0
{\color {Blue} or \; \; \;  \begin{vmatrix}
-2 & 1 & 1 \\ 
1 & -2 & 1 \\ 
1 & 1 &  -2
\end {vmatrix}\begin{bmatrix}
x_1\\ x_2
\\x_3 

\end{bmatrix}= \begin{bmatrix}
0\\ 0
\\0

\end{bmatrix}} 
{\color {Blue} or \; \; \;  \begin{vmatrix}
1 & 1 & -2 \\ 
1 & -2 & 1 \\ 
-2 & 1 &  1
\end {vmatrix}\begin{bmatrix}
x_1\\ x_2
\\x_3 

\end{bmatrix}= \begin{bmatrix}
0\\ 0
\\0

\end{bmatrix}} {\color{Red}\; \; on \; R_ 1 \leftrightarrow  R_ 3}
{\color {Blue} or \; \; \;  \begin{vmatrix}
1 & 1 & -2 \\ 
0 & -3 & 3 \\ 
0 & 0 &  0
\end {vmatrix}\begin{bmatrix}
x_1\\ x_2
\\x_3 

\end{bmatrix}= \begin{bmatrix}
0\\ 0
\\0

\end{bmatrix}} {\color{Red}\; \; on \; R_ 2 \rightarrow R_ 2 - R_1 \; ,\;  R_ 3 \rightarrow R_ 3 + R_2 +R_1}

Rank of the coefficient matrix is 2 and so these equation have 3-2=1 independent solution corresponding to the value  λ=3

These \; Equations \;reduces \;to \\ \\
x_1+x_2+x_3=0 \\
-3x_2+3x_3=0
{\color{Teal}  Solving \; these \; equations \; we \; get \ x_1=x_2=x_3=k

} \\ \\ \therefore  X=\begin{bmatrix}
k\\ k
\\ k

\end{bmatrix}, where \;k\ne0
 X=\begin{bmatrix}
1\\ 1
\\ 1

\end{bmatrix}, is \; the \; characteristic\; vector \; on \; taking \; k =1

Hence characteristic vectors are

 \begin {bmatrix}
1 \\ 0 
\\ -1 

\end {bmatrix},  \begin{bmatrix}
0 \\ 1 
\\ -1 

\end {bmatrix},\begin{bmatrix}
1\\ 1
\\ 1

\end{bmatrix}

Keywords

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