Introduction : We shall find eigen values of the symmetric matrix. Eigen values of a matrix are those numbers that satisfies its characteristic equation.
Question : Find eigen values of the symmetric matrix
\[\left(\begin{matrix}10&-3\\-3&8\\\end{matrix}\right)\]
Solution :
Let
\[A=\left(\begin{matrix}10&-3\\-3&8\\\end{matrix}\right)\]
To find eigen values of A
We shall determine characteristic polynomial of A that is the determinant of A-λI
\[\left|A-\lambda I\right|\] Where I is an identity matrix of order 2. \[\left|\left(\begin{matrix}10&-3\\-3&8\\\end{matrix}\right)-\lambda\left(\begin{matrix}1&0\\0&1\\\end{matrix}\right)\right|\]
\[\left|\left(\begin{matrix}10&-3\\-3&8\\\end{matrix}\right)-\left(\begin{matrix}\lambda&0\\0&\lambda\\\end{matrix}\right)\right|\]
\[\left|\left(\begin{matrix}10-\lambda&-3\\-3&8-\lambda\\\end{matrix}\right)\right|\]
\[\left(10-\lambda\right)\left(8-\lambda\right)-9\]
\[71-18\lambda+\lambda^2\]
So the characteristic equation of A becomes
\[\left|A-\lambda I\right|=0\]
\[71-18\lambda+\lambda^2=0\]
\[\lambda=9-\sqrt{10},\ \ 9+\sqrt{10}\]
Thus we roots of characteristic equation are 9-√10, 9+√10
Hence eigen values of the given matrix A are 9-√10, 9+√10 .