Question

Find the eigenvectors and eigenvalues for the matrix. A =
[2 1
-1 2]

Answer 1

Explanation:

Given matrix A is 2x2

A-kI =0 gives

`\left[\begin{array}{ccc}2-k&1\\-1&2-k\end{array}\right] =0\\\\(2-k)^2+1=0\\2-k=i,-i\\k =2-i, 2+i`

Eigen values are 2-i, 2+i

a) k =2-i

`\left[\begin{array}{ccc}i&1\\-1&i\end{array}\right] =0\\`

i.e. x1=ix2

OR Eigen vector is (i,1)

b) k =2+i

`\left[\begin{array}{ccc}-i&1\\-1&-i\end{array}\right] =0\\`

i.e. x1=-ix2

OR Eigen vector is (-i,1)