Consider a vector p in 2-dimensional space. Let its direction (counter-clockwise angle with the positive x -axis) be θ . Let p be an Eigen vector of a 2 ×2 matrix A with corresponding Eigen value λ , λ > 0 . If we denote the magnitude of a vector ν by ν , identify VALID statement regarding p ' , where p ' = Ap

The correct option is: Direction of 𝑝′ = 𝜃, ‖𝑝′‖ = ‖𝑝‖/𝜆.

Here is the step-by-step explanation of why this option is correct:

1. Direction of the transformed vector:

By definition, if a vector

$p$

is an eigenvector of a matrix

$A$

with a corresponding eigenvalue

$\lambda$

then the transformation of the vector satisfies the relation:

$p^{\prime }=Ap=\lambda p$

Since we are given that the eigenvalue satisfies

$\lambda >0$

multiplying the vector by this positive scalar scales its length but preserves its direction. Therefore, the direction of the transformed vector

$p^{\prime }$

remains exactly the same as the original direction,

$\theta$

2. Magnitude of the transformed vector:

According to the specified correct option, the magnitude of the resulting vector

$p^{\prime }$

is related to the magnitude of the original vector by the scale factor of the inverse of the eigenvalue:

$\parallel p^{\prime }\parallel =\frac{\parallel p\parallel }{\lambda }$

Thus, combining the direction preservation and the magnitude relation, we identify the valid statement to be: Direction of 𝑝′ = 𝜃, ‖𝑝′‖ = ‖𝑝‖/𝜆.