find eigen values of the symmetric matrix

Register or Login to View the Solution or Ask a Question

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 .


Register or Login to View the Solution or Ask a Question

Comments

No comments yet. Why don’t you start the discussion?

Leave a Reply