When we look at the eigen vectors of a matrix , it actually gives us a vector that doesnt change the direction when the matrix is applied … (Ofcourse it is allowed to rotate 180 degrees ….) . At the same time the vector is shrunk to an extent .

If B is the matrix and v is the eigen vector with an eigen value k.

B * v = k*v

if k < 0 then its rotated by 180 degrees. the Eigen vector is shrunk by k.

If B is a symetric matrix then it has linearly independent eigen vectors.