Answer to Question #314303 in Linear Algebra for JKK

Question

Find all Eigen values and corresponding Eigen vectors for the matrix A=

0 0 3

2 5 0

2 3 0

Answer 1

$A=\begin{pmatrix} 0 & 0&3 \\ 2 & 5&0\\ 2&3&0 \end{pmatrix}\\ |A-\lambda I|=\begin{vmatrix} 0-\lambda & 0&3 \\ 2 & 5-\lambda&0\\ 2&3&0-\lambda \end{vmatrix}=\\ =\lambda^2(5-\lambda)+0+18-6(5-\lambda)-0-0=\\ =-\lambda^3+5\lambda^2+6\lambda-12=0\\ \lambda^3-5\lambda^2-6\lambda+12=0\\ 12:\pm1, \pm2,\pm3,\pm4,\pm6,\pm12\\ \lambda=\pm1: (\pm1)^3-5(\pm1)^2-6(\pm1)+12\neq0\\ \lambda=\pm2:(\pm2)^3-5(\pm2)^2-6(\pm2)+12\neq0\\ \lambda=\pm3: (\pm3)^3-5(\pm3)^2-6(\pm3)+12\neq0\\ \lambda=\pm4: (\pm4)^3-5(\pm4)^2-6(\pm4)+12\neq0\\ \lambda=\pm6: (\pm6)^3-5(\pm6)^2-6(\pm6)+12\neq0\\ \lambda=\pm12: (\pm12)^3-5(\pm12)^2-6(\pm12)+12\neq0$

$\lambda_1\approx -2\\ \lambda_2\approx1\\ \lambda_3\approx6$

All Eigen values

$(A-\lambda I)\vec{x}=\vec{0}, \vec{x}\neq\vec{0}\\ 1. \lambda_1=-2\\ \vec{x_1}=(x_1,x_2,x_3)\neq\vec{0}\\ (A-\lambda_1I)\vec{x_1}=\vec{0}\\ \begin{pmatrix} 2 &0&3 \\ 2 & 7&0\\ 2&3&2 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2\\ x_3 \end{pmatrix}=\begin{pmatrix} 0\\0\\0 \end{pmatrix}\\ 2x_1+3x_3=0\\ 2x_1+7x_2=0\\ 2x_1+3x_2+2x_3=0\\ x_3=-\frac{2}{3}x_1\\ x_2=-\frac{2}{7}x_1\\ x_1=1, x_2=-\frac{2}{7}, x_3=-\frac{2}{3}\\ \vec{x_1}=(1,-\frac{2}{7}, -\frac{2}{3})$

$2. \lambda_2=1\\ \vec{x_2}=(x_1,x_2,x_3)\neq\vec{0}\\ (A-\lambda_2I)\vec{x_2}=\vec{0}\\ \begin{pmatrix} -1 &0&3 \\ 2 & 4&0\\ 2&3&-1 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2\\ x_3 \end{pmatrix}=\begin{pmatrix} 0\\0\\0 \end{pmatrix}\\ -x_1+3x_3=0\\ 2x_1+4x_2=0\\ 2x_1+3x_2-x_3=0\\ x_3=\frac{1}{3}x_1\\ x_2=-\frac{1}{2}x_1\\ x_1=1, x_2=-\frac{1}{2}, x_3=\frac{1}{3}\\ \vec{x_2}=(1,-\frac{1}{2}, \frac{1}{3})$

$3. \lambda_3=6\\ \vec{x_3}=(x_1,x_2,x_3)\neq\vec{0}\\ (A-\lambda_3I)\vec{x_3}=\vec{0}\\ \begin{pmatrix} -6 &0&3 \\ 2 & -1&0\\ 2&3&-6 \end{pmatrix}\begin{pmatrix} x_1 \\ x_2\\ x_3 \end{pmatrix}=\begin{pmatrix} 0\\0\\0 \end{pmatrix}\\ -6x_1+3x_3=0\\ 2x_1-x_2=0\\ 2x_1+3x_2-6x_3=0\\ x_3=2x_1\\ x_2=2x_1\\ x_1=1, x_2=2, x_3=2\\ \vec{x_3}=(1,2,2)$

$\vec{x_1}=(1,-\frac{2}{7}, -\frac{2}{3}),\vec{x_2}=(1,-\frac{1}{2}, \frac{1}{3}), \vec{x_3}=(1,2,2)$

All corresponding Eigen vectors