A = ( 0 0 3 2 5 0 2 3 0 ) ∣ A − λ I ∣ = ∣ 0 − λ 0 3 2 5 − λ 0 2 3 0 − λ ∣ = = λ 2 ( 5 − λ ) + 0 + 18 − 6 ( 5 − λ ) − 0 − 0 = = − λ 3 + 5 λ 2 + 6 λ − 12 = 0 λ 3 − 5 λ 2 − 6 λ + 12 = 0 12 : ± 1 , ± 2 , ± 3 , ± 4 , ± 6 , ± 12 λ = ± 1 : ( ± 1 ) 3 − 5 ( ± 1 ) 2 − 6 ( ± 1 ) + 12 ≠ 0 λ = ± 2 : ( ± 2 ) 3 − 5 ( ± 2 ) 2 − 6 ( ± 2 ) + 12 ≠ 0 λ = ± 3 : ( ± 3 ) 3 − 5 ( ± 3 ) 2 − 6 ( ± 3 ) + 12 ≠ 0 λ = ± 4 : ( ± 4 ) 3 − 5 ( ± 4 ) 2 − 6 ( ± 4 ) + 12 ≠ 0 λ = ± 6 : ( ± 6 ) 3 − 5 ( ± 6 ) 2 − 6 ( ± 6 ) + 12 ≠ 0 λ = ± 12 : ( ± 12 ) 3 − 5 ( ± 12 ) 2 − 6 ( ± 12 ) + 12 ≠ 0 A=\begin{pmatrix}
0 & 0&3 \\
2 & 5&0\\
2&3&0
\end{pmatrix}\\
|A-\lambda I|=\begin{vmatrix}
0-\lambda & 0&3 \\
2 & 5-\lambda&0\\
2&3&0-\lambda
\end{vmatrix}=\\
=\lambda^2(5-\lambda)+0+18-6(5-\lambda)-0-0=\\
=-\lambda^3+5\lambda^2+6\lambda-12=0\\
\lambda^3-5\lambda^2-6\lambda+12=0\\
12:\pm1, \pm2,\pm3,\pm4,\pm6,\pm12\\
\lambda=\pm1: (\pm1)^3-5(\pm1)^2-6(\pm1)+12\neq0\\
\lambda=\pm2:(\pm2)^3-5(\pm2)^2-6(\pm2)+12\neq0\\
\lambda=\pm3: (\pm3)^3-5(\pm3)^2-6(\pm3)+12\neq0\\
\lambda=\pm4: (\pm4)^3-5(\pm4)^2-6(\pm4)+12\neq0\\
\lambda=\pm6: (\pm6)^3-5(\pm6)^2-6(\pm6)+12\neq0\\
\lambda=\pm12: (\pm12)^3-5(\pm12)^2-6(\pm12)+12\neq0 A = ⎝ ⎛ 0 2 2 0 5 3 3 0 0 ⎠ ⎞ ∣ A − λ I ∣ = ∣ ∣ 0 − λ 2 2 0 5 − λ 3 3 0 0 − λ ∣ ∣ = = λ 2 ( 5 − λ ) + 0 + 18 − 6 ( 5 − λ ) − 0 − 0 = = − λ 3 + 5 λ 2 + 6 λ − 12 = 0 λ 3 − 5 λ 2 − 6 λ + 12 = 0 12 : ± 1 , ± 2 , ± 3 , ± 4 , ± 6 , ± 12 λ = ± 1 : ( ± 1 ) 3 − 5 ( ± 1 ) 2 − 6 ( ± 1 ) + 12 = 0 λ = ± 2 : ( ± 2 ) 3 − 5 ( ± 2 ) 2 − 6 ( ± 2 ) + 12 = 0 λ = ± 3 : ( ± 3 ) 3 − 5 ( ± 3 ) 2 − 6 ( ± 3 ) + 12 = 0 λ = ± 4 : ( ± 4 ) 3 − 5 ( ± 4 ) 2 − 6 ( ± 4 ) + 12 = 0 λ = ± 6 : ( ± 6 ) 3 − 5 ( ± 6 ) 2 − 6 ( ± 6 ) + 12 = 0 λ = ± 12 : ( ± 12 ) 3 − 5 ( ± 12 ) 2 − 6 ( ± 12 ) + 12 = 0
λ 1 ≈ − 2 λ 2 ≈ 1 λ 3 ≈ 6 \lambda_1\approx -2\\
\lambda_2\approx1\\
\lambda_3\approx6 λ 1 ≈ − 2 λ 2 ≈ 1 λ 3 ≈ 6
All Eigen values
( A − λ I ) x ⃗ = 0 ⃗ , x ⃗ ≠ 0 ⃗ 1. λ 1 = − 2 x 1 ⃗ = ( x 1 , x 2 , x 3 ) ≠ 0 ⃗ ( A − λ 1 I ) x 1 ⃗ = 0 ⃗ ( 2 0 3 2 7 0 2 3 2 ) ( x 1 x 2 x 3 ) = ( 0 0 0 ) 2 x 1 + 3 x 3 = 0 2 x 1 + 7 x 2 = 0 2 x 1 + 3 x 2 + 2 x 3 = 0 x 3 = − 2 3 x 1 x 2 = − 2 7 x 1 x 1 = 1 , x 2 = − 2 7 , x 3 = − 2 3 x 1 ⃗ = ( 1 , − 2 7 , − 2 3 ) (A-\lambda I)\vec{x}=\vec{0}, \vec{x}\neq\vec{0}\\
1. \lambda_1=-2\\
\vec{x_1}=(x_1,x_2,x_3)\neq\vec{0}\\
(A-\lambda_1I)\vec{x_1}=\vec{0}\\
\begin{pmatrix}
2 &0&3 \\
2 & 7&0\\
2&3&2
\end{pmatrix}\begin{pmatrix}
x_1 \\
x_2\\
x_3
\end{pmatrix}=\begin{pmatrix}
0\\0\\0
\end{pmatrix}\\
2x_1+3x_3=0\\
2x_1+7x_2=0\\
2x_1+3x_2+2x_3=0\\
x_3=-\frac{2}{3}x_1\\
x_2=-\frac{2}{7}x_1\\
x_1=1, x_2=-\frac{2}{7}, x_3=-\frac{2}{3}\\
\vec{x_1}=(1,-\frac{2}{7}, -\frac{2}{3}) ( A − λ I ) x = 0 , x = 0 1. λ 1 = − 2 x 1 = ( x 1 , x 2 , x 3 ) = 0 ( A − λ 1 I ) x 1 = 0 ⎝ ⎛ 2 2 2 0 7 3 3 0 2 ⎠ ⎞ ⎝ ⎛ x 1 x 2 x 3 ⎠ ⎞ = ⎝ ⎛ 0 0 0 ⎠ ⎞ 2 x 1 + 3 x 3 = 0 2 x 1 + 7 x 2 = 0 2 x 1 + 3 x 2 + 2 x 3 = 0 x 3 = − 3 2 x 1 x 2 = − 7 2 x 1 x 1 = 1 , x 2 = − 7 2 , x 3 = − 3 2 x 1 = ( 1 , − 7 2 , − 3 2 )
2. λ 2 = 1 x 2 ⃗ = ( x 1 , x 2 , x 3 ) ≠ 0 ⃗ ( A − λ 2 I ) x 2 ⃗ = 0 ⃗ ( − 1 0 3 2 4 0 2 3 − 1 ) ( x 1 x 2 x 3 ) = ( 0 0 0 ) − x 1 + 3 x 3 = 0 2 x 1 + 4 x 2 = 0 2 x 1 + 3 x 2 − x 3 = 0 x 3 = 1 3 x 1 x 2 = − 1 2 x 1 x 1 = 1 , x 2 = − 1 2 , x 3 = 1 3 x 2 ⃗ = ( 1 , − 1 2 , 1 3 ) 2. \lambda_2=1\\
\vec{x_2}=(x_1,x_2,x_3)\neq\vec{0}\\
(A-\lambda_2I)\vec{x_2}=\vec{0}\\
\begin{pmatrix}
-1 &0&3 \\
2 & 4&0\\
2&3&-1
\end{pmatrix}\begin{pmatrix}
x_1 \\
x_2\\
x_3
\end{pmatrix}=\begin{pmatrix}
0\\0\\0
\end{pmatrix}\\
-x_1+3x_3=0\\
2x_1+4x_2=0\\
2x_1+3x_2-x_3=0\\
x_3=\frac{1}{3}x_1\\
x_2=-\frac{1}{2}x_1\\
x_1=1, x_2=-\frac{1}{2}, x_3=\frac{1}{3}\\
\vec{x_2}=(1,-\frac{1}{2}, \frac{1}{3}) 2. λ 2 = 1 x 2 = ( x 1 , x 2 , x 3 ) = 0 ( A − λ 2 I ) x 2 = 0 ⎝ ⎛ − 1 2 2 0 4 3 3 0 − 1 ⎠ ⎞ ⎝ ⎛ x 1 x 2 x 3 ⎠ ⎞ = ⎝ ⎛ 0 0 0 ⎠ ⎞ − x 1 + 3 x 3 = 0 2 x 1 + 4 x 2 = 0 2 x 1 + 3 x 2 − x 3 = 0 x 3 = 3 1 x 1 x 2 = − 2 1 x 1 x 1 = 1 , x 2 = − 2 1 , x 3 = 3 1 x 2 = ( 1 , − 2 1 , 3 1 )
3. λ 3 = 6 x 3 ⃗ = ( x 1 , x 2 , x 3 ) ≠ 0 ⃗ ( A − λ 3 I ) x 3 ⃗ = 0 ⃗ ( − 6 0 3 2 − 1 0 2 3 − 6 ) ( x 1 x 2 x 3 ) = ( 0 0 0 ) − 6 x 1 + 3 x 3 = 0 2 x 1 − x 2 = 0 2 x 1 + 3 x 2 − 6 x 3 = 0 x 3 = 2 x 1 x 2 = 2 x 1 x 1 = 1 , x 2 = 2 , x 3 = 2 x 3 ⃗ = ( 1 , 2 , 2 ) 3. \lambda_3=6\\
\vec{x_3}=(x_1,x_2,x_3)\neq\vec{0}\\
(A-\lambda_3I)\vec{x_3}=\vec{0}\\
\begin{pmatrix}
-6 &0&3 \\
2 & -1&0\\
2&3&-6
\end{pmatrix}\begin{pmatrix}
x_1 \\
x_2\\
x_3
\end{pmatrix}=\begin{pmatrix}
0\\0\\0
\end{pmatrix}\\
-6x_1+3x_3=0\\
2x_1-x_2=0\\
2x_1+3x_2-6x_3=0\\
x_3=2x_1\\
x_2=2x_1\\
x_1=1, x_2=2, x_3=2\\
\vec{x_3}=(1,2,2) 3. λ 3 = 6 x 3 = ( x 1 , x 2 , x 3 ) = 0 ( A − λ 3 I ) x 3 = 0 ⎝ ⎛ − 6 2 2 0 − 1 3 3 0 − 6 ⎠ ⎞ ⎝ ⎛ x 1 x 2 x 3 ⎠ ⎞ = ⎝ ⎛ 0 0 0 ⎠ ⎞ − 6 x 1 + 3 x 3 = 0 2 x 1 − x 2 = 0 2 x 1 + 3 x 2 − 6 x 3 = 0 x 3 = 2 x 1 x 2 = 2 x 1 x 1 = 1 , x 2 = 2 , x 3 = 2 x 3 = ( 1 , 2 , 2 )
x 1 ⃗ = ( 1 , − 2 7 , − 2 3 ) , x 2 ⃗ = ( 1 , − 1 2 , 1 3 ) , x 3 ⃗ = ( 1 , 2 , 2 ) \vec{x_1}=(1,-\frac{2}{7}, -\frac{2}{3}),\vec{x_2}=(1,-\frac{1}{2}, \frac{1}{3}),
\vec{x_3}=(1,2,2) x 1 = ( 1 , − 7 2 , − 3 2 ) , x 2 = ( 1 , − 2 1 , 3 1 ) , x 3 = ( 1 , 2 , 2 )
All corresponding Eigen vectors
#332078 on Oct 2022
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