A=⎝⎛022053300⎠⎞∣A−λI∣=∣∣0−λ2205−λ3300−λ∣∣==λ2(5−λ)+0+18−6(5−λ)−0−0==−λ3+5λ2+6λ−12=0λ3−5λ2−6λ+12=012:±1,±2,±3,±4,±6,±12λ=±1:(±1)3−5(±1)2−6(±1)+12=0λ=±2:(±2)3−5(±2)2−6(±2)+12=0λ=±3:(±3)3−5(±3)2−6(±3)+12=0λ=±4:(±4)3−5(±4)2−6(±4)+12=0λ=±6:(±6)3−5(±6)2−6(±6)+12=0λ=±12:(±12)3−5(±12)2−6(±12)+12=0

λ1≈−2λ2≈1λ3≈6
All Eigen values
(A−λI)x=0,x=01.λ1=−2x1=(x1,x2,x3)=0(A−λ1I)x1=0⎝⎛222073302⎠⎞⎝⎛x1x2x3⎠⎞=⎝⎛000⎠⎞2x1+3x3=02x1+7x2=02x1+3x2+2x3=0x3=−32x1x2=−72x1x1=1,x2=−72,x3=−32x1=(1,−72,−32)
2.λ2=1x2=(x1,x2,x3)=0(A−λ2I)x2=0⎝⎛−12204330−1⎠⎞⎝⎛x1x2x3⎠⎞=⎝⎛000⎠⎞−x1+3x3=02x1+4x2=02x1+3x2−x3=0x3=31x1x2=−21x1x1=1,x2=−21,x3=31x2=(1,−21,31)
3.λ3=6x3=(x1,x2,x3)=0(A−λ3I)x3=0⎝⎛−6220−1330−6⎠⎞⎝⎛x1x2x3⎠⎞=⎝⎛000⎠⎞−6x1+3x3=02x1−x2=02x1+3x2−6x3=0x3=2x1x2=2x1x1=1,x2=2,x3=2x3=(1,2,2)
x1=(1,−72,−32),x2=(1,−21,31),x3=(1,2,2)
All corresponding Eigen vectors
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